Appendix

Boundary Curvatures and the Distance Function

Let Q be a domain in !Rn having non-empty boundary t3Q. The distance function d is defined by

(A 1) d(x)=dist (x, oQ).

It is readily shown that dis uniformly Lipschitz continuous. For let x, y E !Rn and choose z E oQ such that I y- zl = d(y). Then

d(x)~lx-zl ~lx-yl +d(y)

so that by interchanging x andy we have

(A 2) ld(x)-d(y)l ~lx-Yl·

Now let oQ E C2 . For y E oQ, let v(y), and T(y) denote respectively the unit inner normal to oQ at y and the tangent hyperplane to oQ at y. The curvatures of oQ at a fixed point Yo E oQ are determined as follows. By a rotation of coordinates we can assume that the xn coordinate axis lies in the direction v(y0). In some neighborhood %=%(yo) of Yo, oQ is then given by Xn=<p(x') where x'= (x 1, ... , xn_ 1), <p E C 2(T(y0 ) n %) and D<p(y~) =0. The curvature of oQ at Yo is then described by the orthogonal invariants of the Hessian matrix [D2 <p] evaluated aty~. Theeigenvaluesof[D2<p(y~)], K 1, ... , Kn_ 1, are called the principal curvatures of oQ at y 0 and the corresponding eigenvectors are called the principal directions of oQ at Yo· The mean curvature of oQ at Yo is given by

(A 3)

By a further rotation of coordinates we can assume that the x 1, ... , xn_ 1 axes lie along principal directions corresponding to Kp . .. , "n- 1 at y0 . Let us call such a coordinate system a principal coordinate system at y 0 • The Hessian matrix [D2<p(y~)] with respect to the principal coordinate system at y 0 described above is given by

382 Appendix: Boundary Curvatures and the Distance Function

The unit inner normal vector v(y')=v(y) at the point y=(y', <p(y')) E JV n oQ is given by

(A 5) -D;<p(y') 1

v;(y) , i=1, ... ,n-1, vnCy) J1 +ID<p(y')l 2 ~1 + ID<p(y')l 2

Hence with respect to the principal coordinate system at y 0 , we have

For 11>0, let us set T 11 = {x E Q I d(x)<f.l}. The following lemma relates the smoothness ofthe distance function din rJl to that ofthe boundary oQ.

Lemma 1. Let Q be bounded and oQ E Ck for k~2. Then there exists a positive constant f.1 depending on Q such that dE Ck(T 11 ).

Proof The conditions on Q imply that oQ satisfies a uniform interior sphere condition; that is, at each point y 0 E oQ there exists a ball B depending on y 0 such that B n {!ffi"-Q)= y 0 and the radii of the balls Bare bounded from below by a positive constant, which we take to be f.l· It is easy to show that f.l- 1 bounds the principal curvatures of oQ. Also, for each point X E rJl, there will exist a unique pointy= y(x) E oQ such that lx-yl =d(x). The points x andy are related by

(A7) x=y+v(y)d.

We show that the equation (A 7) determines y and d as ck- 1 functions of X. For a fixed point x 0 E r 11 , let y 0 = y(x0 ) and choose a principal coordinate system at y 0 •

We define a mapping g={g 1 , .•• , g") from Olt=(T(y0 ) n .JV(y0 )) x lffi into lffi" by

g(y', d)=y+v(y)d, y=(y', <p(y')).

Clearly g E ck- 1(0U), and the Jacobian matrix of gat (y~, d(x)) is given by

(A8) [Dg]=diag[1-KA ... , 1-Kn-A 1].

Since the Jacobian of gat (y~, d(x 0)) is given by

it follows from the inverse mapping theorem that for some neighborhood "'tft = ~(x0), the mapping y' is contained in ck- 1(~). From (A 7) we have Dd(x)= v(y(x))=v(y'(x)) E ck- 1 (~)for X E ~. Henced E Ck(Jt),and thusd E Ck(TJl). 0

An expression for the Hessian matrix of d at points close to oQ is an immediate consequence of the proof of Lemma 1.

Appendix: Boundary Curvatures and the Distance Function 383

Lemma 2. Let Q and Jl satisfy the conditions of Lemma I and let Xo E r I'' Yo E oQ be such that lx0 - y01 = d(x0). Then, in terms of a principal coordinate system at y 0 , we have

[D2-l( J d. [ -Kl -Kn-l 0~ (A 10) w Xo) = mg 1-Kld' .. . , 1-Kn-ld' J

Proof Since

Dd(x0)=v(y0)=(0, 0, ... , 1)

we have D;n(x0)=0, i= I, ... , n. To obtain the other derivatives we write, for i,j= I, ... , n-1,

by (A 6) and (A 8). D

Note that the result of Lemma 2 is equivalent to the geometrically evident statement that the circles of principal curvature to oQ at y 0 and to the parallel surface through x 0 at x 0 are concentric.

Mean Curvature

Let us derive now a formula for the mean curvature of a C 2 hypersurface 6 in terms of its given representation. Let y 0 E 6 and suppose that in a neighborhood .AI' of y 0 , 6 is given by lj;(x)=O where ljJ E C 2(JV) and IDl/11 >0 in .AI. The unit normal to 6 at a pointy E 6 n .AI (directed towards positive lj;) is given by

(A II)

Let Kp ... , Kn-l be the principal curvatures of 6 at Yo- Then with respect to a corresponding principal coordinate system at y 0 , one can show that

(A 12) i= I, ... , n.

384 Appendix: Boundary Curvatures and the Distance Function

Consequently the eigenvalues of the matrix [D/D;!/1/ID!/11)] at y 0 with respect to the original coordinates are - Kp . •. , - Kn-1' 0 and hence the mean curvature of 6 at Yo is given by

(Al3)

In particular if 6 is the graph in IRin+ 1 of a function of n variables u E C2(Q), that is, 6 is defined by xn + 1 = u(x 1' ... , xn ), the mean curvature of 6 at x 0 E Q is given by

(A 14) 10 [ D.u ]) H(x 0 ) =- D; ' . n j1 +IDul 2 x=xa

6 is called a minimal surface if H(x 0 ) = 0 for all x 0 for all x 0 E Q. Note that we also obtain from (A 12) the following formula for the sum of the

squares of the principal curvatures K 1 , ... , Kn at x 0 ,

n n

(A 15) rc 2 = I K~(xo)= I DivjDjv;(xo), i=1 i,j=1

where

i= 1, ... , n.

Bibliography

Adams, R. A. [AD] Sobolev Spaces. New York: Academic Press 1975.

Agmon, S. [AG] Lectures on Elliptic Boundary Value Problems. Princeton, N.J.: Van Nostrand 1965.

Agmon, S., A. Douglis and L. Nirenberg [AD N I] Estimates near the boundary for solutions of elliptic partial differential equations satisfying

general boundary conditions. I. Comm. Pure Appl. Math. 12, 623-727 (1959). [ADN 2] Estimates near the boundary for solutions of elliptic partial differential equations satisfying

general boundary conditions. II. Comm. Pure Appl. Math. 17, 35--92 (1964).

Aleksandrov, A. D. [AL I] Majorization of solutions of second-order linear equations. Vestnik Leningrad. Univ.

21, no. I, 5-25 (1966) [Russian]. English Translation in Amer. Math. Soc. Trans!. (2) 68, 120-143 (1968).

[AL 2] Majorants of solutions and uniqueness conditions for elliptic equations. Vestnik Lenin­grad. Univ. 21, no. 7, 5-20 (1966). English Translation in Amer. Math. Soc. Trans!. (2) 68, 144--161 (1968).

Allard, W. [AA] On the first variation of a varifold. Ann. of Math. (2) 95, 417-491 (1972).

Almgren, F. J. [AM] Some interior regularity theorems for minimal surfaces and an extension of Bernstein's

theorem. Ann. of Math. (2) 84, 277-292 ( 1966).

Bakel'man, I. Y A. [BA I] Mean curvature and quasilinear elliptic equations. Sibirsk. Mat. Z. 9, 1014-1040 (1968)

[Russian]. English Translation in Siberian Math. J. 9, 752-771 (1968). [BA 2] Geometric problems in quasilinear elliptic equations. Uspehi Mat. Nauk 25, no. 3, 49-

112 (1970) [Russian]. English Translation in Russian Math. Surveys 25, no. 3, 45-109 (1970).

Bernstein, S. [BE I] Sur Ia generalisation du probleme de Dirichlet. I. Math. Ann. 62, 253-271 (1906). [BE 2] Methode generale pour Ia resolution du probleme de Dirichlet. C. R. Acad. Sci. Paris

[BE 3] [BE4]

[BE 5]

[BE 6]

144, 1025-1027 (1907). Sur Ia generalisation du probleme de Dirichlet II. Math. Ann. 69, 82-136 (1910). Conditions necessaires et suffisantes pour Ia possibilite du probleme de Dirichlet. C. R. Acad. Sci. Paris 150, 514--515 (1910). Sur les surfaces definies au moyen de leur courbure moyenne et totale. Ann. Sci. Ecole Norm. Sup. 27, 233-256 (1910). Surles equations du calcul des variations. Ann. Sci. Ecole Norm. Sup. 29, 431-485 (1912).

Bers, L., and L. Nirenberg [BN] On linear and non-linear elliptic boundary value problems in the plane. In: Convegno

Internazionale sulle Equazioni Lineari aile Derivate Parziali, Trieste, pp. 141-167. Rome: Edizioni Cremonese 1955.

Bers, L., and M. Schechter [BS] Elliptic equations. In: Partial Differential Equations, pp. 131-299. New York: Interscience

1964.

Bliss, G. A. [BL] An integral inequality. J. London Math. Soc. 5, 40-46 (1930).

386 Bibliography

Bohl, P. [BO] Uber die Bewegung eines mechanischen Systems in der Nahe einer Gleichgewichtslage.

J. Reine Angew. Math. 127, 179-276 (1904).

Bombieri, E. [BM] Variational problems and elliptic equations. In: Proceedings of the International Congress

of Mathematicians, Vancouver 1974, Vol. 1, 53---{;3. Vancouver: Canadian Mathematical Congress 197 5.

Bombieri, E., E. De Giorgi and E. Giusti [BDG] Minimal cones and the Bernstein problem. Invent. Math. 7, 243-268 (1969).

Bombieri, E., E. De Giorgi and M. Miranda [BDM] Una maggiorazione a priori relativa aile ipersuperfici minimali non parametriche. Arch.

Rational Mech. Anal. 32, 255-267 (1969).

Bombieri, E., and E. Giusti [BG I] Harnack's inequality for elliptic differential equations on minimal surfaces. Invent. Math.

15, 24--46 (1972). [BG 2] Local estimates for the gradient of non-parametric surfaces of prescribed mean curvature.

Comm. Pure Appl. Math. 26,381-394 (1973).

Bouligand, G., G. Giraud and P. Delens [BGD] Le probleme de Ia derivee oblique en theorie du potentiel. Actualites Scientifiques et

Industrielles 219. Paris: Hermann 1935.

Brandt, A. [BR I] Interior estimates for second-order elliptic differential (or finite-difference) equations via

the maximum principle. Israel J. Math. 7, 95-121 (1969). [BR 2] Interior Schauder estimates for parabolic differential- (or difference-) equations via the

maximum principle. Israel J. Math. 7, 254-262 (1969).

Browder, F. E. [BW 1] Strongly elliptic systems of differential equations. In: Contributions to the Theory of Partial

Differential Equations, pp. 15-51. Princeton, N.J.: Princeton University Press 1954. [BW 2] Problemes non-lineaires. Seminaire de Mathematiques Superieures, No. 15 ('Ete, 1965).

Montreal, Que.: Les Presses de l'Universite de Montreal 1966. [BW 3] Existence theorems for nonlinear partial differential equations. In: Proceedings of Symposia

in Pure Mathematics, Volume XVI; pp. J---{;0. Providence, R.I.: American Mathematical Society 1970.

Caccioppoli, R. [CA I] Sulle equazioni ellittiche a derivate parziali con n variabili indipendenti. Atti Accad.

Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 19, 83-89 (1934). [CA 2] Sulle equazioni ellittiche a derivate parziali con due variabili indipendenti, e sui problemi

regolari di calcolo delle variazioni. I. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 22, 305-310 (1935).

[CA 3] Sulle equazioni ellittiche a derivate parziali con due variabili indipendenti, e sui problemi regolari di calcolo delle variazioni. II. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 22, 376-379 (1935).

[CA 4] Limitazioni integrali perle soluzioni di un'equazione lineare ellittica a derivate parziali. Giorn. Mat. Battaglini (4) 4 (80), 186-212 (1951).

Calderon, A. P., and A. Zygmund [CZ] On the existence of certain singular integrals. Acta Math. 88, 85-139 (1952).

Campanato, S. [CM I] Proprietit di inclusione per spazi di Morrey. Ricerche Mat. 12, 67-86 (1963). [CM 2] Equazioni ellittiche del Il 0 ordine e spazi !1'(2 ·•>. Ann. Mat. Pura Appl. (4) 69, 321-381

(1965).

Chicco, M. [CI I] Principia di massimo forte per sottosoluzioni di equazioni ellittiche di tipo variazionale.

Boll. Un. Mat. Ita!. (3) 22, 368-372 (1967). [CI 2] Semicontinuitit delle sottosoluzioni di equazioni ellittiche di tipo variazionale. Boll. Un.

Mat. Ita!. (4) 1, 548-553 (1968). [CI 3] Principia di massimo per soluzioni di problemi a! contorno misti per equazioni ellittiche

di tipo variazionale. Boll. Un. Mat. Ita!. (4) 3, 384-394 (1970).

Bibliography 387

[CI 4] Solvability of the Dirichlet problem in H 2·P(Q) for a class of linear second order elliptic partial differential equations. BolL Un. Mat. Ita!. (4) 4, 374--387 (1971).

[CI 5] Sulle equazioni ellittiche del secondo ordine a coefficienti continui. Ann. Mat. Pura Appl. (4) 88, 123-133 (1971).

Cordes, H. 0. [CO I] Uber die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter

Ordnung in mehr als zwei Variablen. Math. Ann. 131, 278-312 (1956). [CO 2] Zero order a priori estimates for solutions of elliptic differential equations. In: Proceedings

of Symposia in Pure Mathematics, Volume IV, pp. 157-166. Providence, R.I.: American Mathematical Society 1961.

Courant, R., and D. Hilbert [CH] Methods of Mathematical Physics. Volume II. New York: Interscience 1962.

De Giorgi, E. [DG I] Sulla differenziabilita e l'analiticita delle estremali degli integrali multipli regolari. Mem.

Accad. Sci. Torino CL Sci. Fis. Mat. Natur. (3) 3, 25-43 (1957). [DG 2] Una estensione del teorema di Bernstein. Ann. Scuola Norm. Sup. Pisa (3) 19, 79-85 (1965).

Douglas, J., T. Dupont and J. Serrin [DDS] Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form.

Arch. Rational Mech. AnaL 42, 157-168 (1971).

Douglis, A. and L. Nirenberg [DN] Interior estimates for elliptic systems of partial differential equations. Comm. Pure AppL

Math. 8, 503-538 (1955).

Dunford, N., and J. T. Schwartz [DS] Linear Operators, Part I. New York: Interscience 1958.

Edmunds, D. E. [ED] Quasilinear second order elliptic and parabolic equations. Bull. London Math. Soc. 2,

5-28 ( 1970).

Edwards, R. E. [EW] Functional Analysis: Theory and Applications. New York: Holt, Rinehart and Winston

1965.

Egorov, Yu. V., and V. A. Kondrat'ev [EK] The oblique derivative problem. Mat. Sb. (N.S.) 78 (120), 148-176 (1969) [Russian].

English Translation in Math USSR Sb. 7, 139-169 (1969). Federer, H. [FE] Geometric Measure Theory. Berlin-Heidelberg-New York: Springer-Verlag 1969.

Finn, R. [FN I] [FN 2] [FN 3]

[FN 4]

A property of minimal surfaces. Proc. Nat. Acad. Sci. U.S.A., 39, 197-201 (1953). On equations of minimal surface type. Ann. of Math. (2) 60, 397-416 (1954). New estimates for equations of minimal surface type. Arch. Rational Mech. Anal. 14, 337-375 (1963). Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature. J. Analyse Math. 14, 139-160 (1965).

Finn, R., and D. Gilbarg [FG I] Asymptotic behavior and uniqueness of plane subsonic flows. Comm. Pure Appl. Math.

10, 23-63 (1957). [FG 2] Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential

equations. Acta Math. 98, 265-296 (1957).

Finn, R., and J. Serrin [FS] On the Holder continuity of quasi-conformal and elliptic mappings. Trans. Amer. Math.

Soc. 89, 1-15 (1958).

Fiorenza, R. [FI I] Sui problemi di derivata obliqua perle equazioni ellittiche. Ricerche Mat. 8, 83-110 (1959). [FI 2] Sui problemi di derivata obliqua perle equazioni ellittiche quasi lineari. Ricerche Mat. 15,

74--108 (1966).

Fleming, W. [FL] On the oriented Plateau problem. Rend. Circ. Mat. Palermo (2) 11, 69-90 (1962).

388 Bibliography

Friedman, A. [FR] Partial Differential Equations. New York: Academic Press 1969.

Friedrichs, K. 0. [FD I] The identity of weak and strong extensions of differential operators. Trans. Amer. Math.

Soc. SS, 132-151 (1944). [FD 2] On the differentiability of the solutions of linear elliptic differential equations. Comm.

Pure Appl. Math. 6, 299-326 (1953).

Giirding, L. [GA] Dirichlet's problem for linear elliptic partial differential equations. Math. Scand. 1, 55-72

(1953).

Gariepy, R., and W. P. Ziemer [GZ] Behaviour at the boundary of solutions of quasilinear elliptic equations. Arch. Rational

Mech. Anal. 56, 372-384 (1972).

Gerhardt, C. [GE] Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean

curvature. Math. Z. 139, 173-198 (1974).

Giaquinta, M. [GI] On the Dirichlet problem for surfaces of prescribed mean curvature. Manuscripta Math.

12, 73-86 (1974).

Gilbarg, D. [GL I] Some local properties of elliptic equations. In: Proceedings of Symposia in Pure Mathe­

matics, Volume IV, pp. 127-141. Providence, R.I.: American Mathematical Society 1961. [GL 2] Boundary value problems for nonlinear elliptic equations in n variables. In: Nonlinear

Problems, pp. 151-159. Madison, Wis.: University of Wisconsin Press 1963.

Gilbarg, D., and J. Serrin [GS] On isolated singularities of solutions of second order elliptic differential equations. J.

Analyse Math. 4, 309-340 (1955/56).

Giraud, G. [GR I] Sur le probleme de Dirichlet generalise (deuxieme memoire). Ann. Sci. Ecole Norm. Sup.

(3) 46, 131-245 (1929). [GR 2] Generalisation des problemes sur les operations du type elliptiques. Bull. Sci. Math. (2)

56, 248-272, 281-312, 316-352 (1932). [GR 3] Sur certains problemes non lineaires de Neumann et sur certains problemes non lineaires

mixtes. Ann. Sci. Ecole Norm. Sup. (3) 49, 1-104, 245-309 (1932).

Giusti, E. [GT 1] [GT2]

Greco, D. [GC]

Superfici cartesiane di area minima. Rend. Sem. Mat. Fis. Milano 40, 3-21 (1970). Boundary value problems for non-parametric surfaces of prescribed mean curvature. Ann. Scuola Norm. Sup. Pisa (4) 3, 501-548 (1976).

Nuove formole integrali di maggiorazione per le soluzioni di un'equazione lineare di tipo ellittico ed applicazioni alia teoria del potenziale. Ricerche Mat. S, 126-149 (1956).

Gunter, N. M. [GU] Potential Theory and its Applications to Basic Problems of Mathematical Physics. New

York: Frederick Ungar Publishing Co. 1967.

Hardy, G. H., and J. E. Littlewood [HL] Some properties of fractional integrals. I. Math. Z. 27, 565-606 (1928).

Hardy, G. H., J. E. Littlewood, and G. Polya [HLP] Inequalities. 2nd ed., Cambridge: Cambridge University Press 1952.

Hartman, P. [HA] On the bounded slope condition. Pacific J. Math. 18, 495-511 (1966).

Hartman, P. and L. Nirenberg [HN] On spherical image maps whose Jacobians do not change sign. Amer. J. Math. 81, 901-

920 (1959).

Hartman, P., and G. Stampacchia [HS] On some non-linear elliptic differential-functional equations. Acta Math. 115, 271-310

(1966).

Bibliography 389

Heinz, E. [HE I]

[HE 2]

Uber die Liisungen der Minimalfliichengleichung. Nachr. Akad. Wiss. Giittingen. Math.­Phys. Kl. Ila, 51-56 (1952). Interior gradient estimates for surfaces z= f(x, y) with prescribed mean curvature. J. Differential Geometry 5, 149-157 (1971).

Herve, R.-M. [HR] Recherches axiomatiques sur Ia theorie des fonctions surharmoniques et du potentiel.

Ann. Inst. Fourier (Grenoble) 12,415-571 (1962).

Herve, R.-M., and M. Herve [HH] Les fonctions surharmoniques associees a un operateur elliptique du second ordre a

coefficients discontinus. Ann. Inst. Fourier (Grenoble) 19, fasc. I, 305-359 (1969).

Hilbert, D. [HI] Uber das Dirichletsche Prinzip. Jber. Deutsch. Math.-Verein. 8, 184-188 (1900).

Hildebrandt, S. [HD] Maximum principles for minimal surfaces and for surfaces of continuous mean curvature.

Hopf, E. [HOI]

[HO 2]

[HO 3]

[H04]

[HO 5]

[HO 6]

Math. Z. 128, 253-269 (1972).

Elementare Bemerkungen tiber die Liisungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitz. Ber. Preuss. Akad. Wissensch. Berlin, Math.-Phys. Kl. 19, 147-152 (1927). Zum analytischen Charakter der Liisungen reguliirer zweidimensionaler Variationspro­bleme. Math. Z. 30, 404-413 (1929). Uber den funktionalen, insbesondere den analytischen Charakter der Liisungen elliptischer Differentialgleichungen zweiter Ordnung. Math. Z. 34, 194-233 (1932). On S. Bernstein's theorem on surfaces z(x, y) ofnonpositive curvature. Proc. Amer. Math. Soc. 1, 80-85 (1950). A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc. 3, 791-793 (1952). On an inequality for minimal surfaces z=z(x, y). J. Rational Mech. Anal. 2, 519-522 (1953).

Hiirmander, L. [HM I] Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. (2)

83, 129-209 (1966). [HM 2] The boundary p~oblems of physical geodesy. Arch. Rational Mech. Anal. 62, 1-52 (1976).

Ivanov, A. V. [IV I] Interior estimates of the first derivatives of the solutions of quasi-linear nonuniformly

elliptic and non uniformly parabolic equations of general form. Zap. Naucn. Sem. Lenin­grad. Otdel. Mat. Inst. Steklov. (LOMI) 14, 24-47 (1969) [Russian]. English Translation in Sem. Math. V. A. Steklov Math. Inst. Leningrad 14, 9-21 (1971).

[IV 2] The Dirichlet problem for second-order quasi-linear nonuniformly elliptic equations. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 19, 79-94(1970) [Russian]. English Translation in Sem. Math. V. A. Steklov Math. Inst. Leningrad 19, 43-52 (1972).

Ivoekina, N. M., and A. P. Oskolkov [IO] N onlocal estimates for the first derivatives of solutions of the first boundary problem for

certain classes ofnonuniformly elliptic and nonuniformly parabolic equations and systems. Trudy Mat. Inst. Steklov 110, 65-101 (1970) [Russian]. English Translation in Proc. Steklov lust. Math. 110, 72-ll5 (1970).

Jenkins, H. [JE] On 2-dimensional variational problems in parametric form. Arch. Rational Mech. Anal.

8, 181-206 (1961).

Jenkins, H., and J. Serrin [JS I] Variational problems of minimal surface type I. Arch. Rational Mech. Anal. 12, 185-212

(1963). [JS 2] The Dirichlet problem for the minimal surface equation in higher dimensions. J. Reine

Angew. Math. 229, 170-187 (1968).

John, F., and L. Nirenberg [JN] On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415-426 (1961).

390 Bibliography

Kellogg, 0. D. [KE 1] On the derivatives of harmonic functions on the boundary. Trans. Amer. Math. Soc. 33,

486-510 (1931). [KE 2] Foundations of Potential Theory. New York: Dover 1954.

Kondrachov, V.I. [KN] Sur certaines proprietes des fonctions dans l'espace U. C. R. (Doklady) Acad. Sci. URSS

(N. S.) 48, 535--538 (1945). Kom,A. [KR 1]

[KR2]

Uber Minimalfliichen, deren Randkurven wenig von ebenen Kurven abweichen. Abhandl. Konig!. Preuss. Akad. Wiss., Berlin 1909; Anhang, Abh. 2. Zwei Anwendungen der Methode der sukzessiven Anniiherungen. Schwarz Festschrift, Berlin 1914, 215-229.

Koselev, A. E. [KO] On boundedness in U of derivatives of solutions of elliptic differential equations. Mat.

Sb. (N. S.) 38 (80), 359-372 (1956) [Russian].

Ladyzhenskaya, 0. A. [LA] Solution of the first boundary problem in the large for quasi-linear parabolic equations.

Trudy Moskov. Mat. Obsc. 7, 149-177 (1958) [Russian].

Ladyzhenskaya, 0. A., and N. N. Ural'tseva [LU 1] On the smoothness of weak solutions of quasilinear equations in several variables and of

variational problems. Comm. Pure Appl. Math. 14,481-495 (1961). [LU 2] Quasi!inear elliptic equations and variational problems with several independent variables.

Uspehi Mat. Nauk 16, no. 1, 19-90 (1961) [Russian]. English Translation in Russian Math. Surveys 16, no. 1, 17-92 (1961).

[L U 3] On Holder continuity of solutions and their derivatives of linear and quasilinear elliptic and parabolic equations. Trudy Mat. Inst. Steklov. 73, 172-220 (1964) [Russian]. English Translation in Amer. Math. Soc. Trans!. (2) 61, 207-269 (1967).

[LU 4] Linear and Quasilinear Elliptic Equations. Moscow: Izdat. "Nauka" 1964 [Russian]. English Translation: New York: Academic Press 1968. 2nd Russian ed. 1973.

[LU 5] Global estimates of the first derivatives of the solutions of quasi-linear elliptic and parabolic equations. Zap. Naucn. Sem. Leningrad. Otdel. Mat. lnst. Steklov. (LOMI) 14, 127-155 (1969) [Russian]. English Translation in Sem. Math. V. A. Steklov Math. Inst. Leningrad 14, 63-77 (1971).

[LU 6] Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Comm. Pure Appl. Math. 23, 677-703 (1970).

Landis, E. M. [LN] Harnack's inequality for second order elliptic equations of Cordes type. Dokl. Akad.

Lax, P. D.

Nauk SSSR 179, 1272-1275 (1968) [Russian]. English Translation in Soviet Math. Dokl. 9, 540-543 (1968).

[LX] On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations. Comm. Pure Appl. Maths. 8, 615-633 (1955).

Lax, P. D., and A. M. Milgram [LM] Parabolic equations. In: Contributions to the Theory of Partial Differential Equations,

pp. 167-190. Princeton, N.J.: Princeton University Press 1954.

Lebesgue, H. [LE] Sur le probleme de Dirichlet. Rend. Circ. Mat. Palermo 24, 371-402 (1907).

Leray, J. [LR] Discussion d'un probleme de Dirichlet. J. Math. Pures Appl. 18, 249-284 (1939).

Leray, J., et J.-L. Lions [LL] Quelques resultats de Visik sur les problemes elliptiques non lineaires par les methodes de

Minty-Browder. Bull. Soc. Math. France 93, 97-107 (1965).

Leray, J., et J. Schauder [LS] Topologie et equations fonctionelles. Ann. Sci. Ecole Norm. Sup. 51, 45-78 (1934).

Lewy, H., and G. Stampacchia [LST] On existence and smoothness of solutions of some noncoercive variational inequalities.

Arch. Rational Mech. Anal. 41, 241-253 (1971).

Bibliography 391

Lichtenstein, L. [LC] Neuere Entwicklung der Theorie partieller Differentialgleichungen zweiter Ordnung

vom elliptischen Typus. In: En c. d. Math. Wissensch. 2.3.2, 1277-1334 (1924 ). Leipzig: B. G. Teubner 1923-27.

Littman, W., G. Stampacchia and H. F. Weinberger [LSW] Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm.

Sup. Pisa (3) 17, 43-77 (1963).

Meyers, N. G. [ME I] An LP-estimate for the gradient of solutions of second order elliptic divergence equations.

Ann. Scuola Norm. Sup. Pisa (3) 17, 189-206 (1963). [ME 2] An example of non-uniqueness in the theory of quasilinear elliptic equations of second order.

Arch. Rational Mech. Anal. 14, 177-179 (1963).

Meyers, N. G., and J. Serrin [MS I] The exterior Dirichlet problem for second order elliptic partial differential equations.

J. Math. Mech. 9, 513-538 (1960). [MS 2] H= W. Proc. Nat. Acad. Sci. U.S.A. 51, 1055-1056 (1964).

Michael, J. H. [MI] A general theory for linear elliptic partial differential equations. J. Differential Equations

23, 1-29 (1977).

Michael, J. H., and L. M. Simon [MSI] Sobolev and mean-value inequalities on generalized submanifolds of R". Comm. Pure

Appl. Math. 26, 361-379 (1973).

Miller, K. [ML 1]

[ML2]

[ML 3]

[ML4]

Barriers on cones for uniformly elliptic operators. Ann. Mat. Pura Appl. (4) 76, 93-105 (1967). Exceptional boundary points for the nondivergence equation which are regular for the Laplace equation and vice-versa. Ann. Scuola Norm. Sup. Pisa (3) 22, 315-330 (1968). Extremal barriers on cones with Phragm(m-Lindelof theorems and other applications. Ann. Mat. Pura Appl. (4) 90, 297-329 (1971). Nonequivalence of regular boundary points for the Laplace and nondivergence equations, even with continuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 24, 159-163 (1970).

Miranda, C. [MR I] Sui problema misto perle equazioni lineari ellittiche. Ann. Mat. Pura Appl. (4) 39, 279-

303 (1955). [MR 2] Partial Differential Equations of Elliptic Type. 2nd ed., Berlin-Heidelberg-New York:

Springer-Verlag 1970.

Miranda, M. [MD I] Diseguaglianze di Sobolev sulle ipersuperfici minimali. Rend. Sem. Math. Univ. Padova

38, 69-79 (1967). [MD 2] Una maggiorazione integrale per le curvature delle ipersuperfici minimale. Rend. Sem.

Mat. Univ. Padova 38, 91-107 (1967). [MD 3] Comportamento delle successioni convergenti di frontiere minimali. Rend. Sem. Mat.

Univ. Padova 38, 238-257 (1967). [MD 4] Un principio di massimo forte per le frontiere minimali e una sua applicazione alia riso­

luzione del problema al contomo per l'equazione delle superfici di area minima. Rend. Sem. Mat. Univ. Padova 45, 355-366 (1971).

[MD 5] Dirichlet problem with L 1 data for the non-homogeneous minimal surface equation. Indiana Univ. Math. J. 24, 227-241 (1974).

Morrey, C. B., Jr. [MY I] On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math.

Soc. 43, 126-166 (1938). [MY 2] Functions of several variables and absolute continuity, II. Duke Math. J. 6, 187-215

(1940). [MY 3] Multiple integral problems in the calculus of variations and related topics. Univ. California

Pub!. Math. (N. S.) 1, 1-130 (1943). [MY 4] Second order elliptic equations in several variables and Holder continuity. Math. Z. 72,

146-164 (1959).

392 Bibliography

[MY 5] Multiple Integrals in the Calculus of Variations. Berlin-Heidelberg-New York: Springer­Verlag 1966.

Moser, J. [MJ 1]

[MJ 2]

Nash, J. [NA]

Neeas, J. [NE]

A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13,457-468 (1960). On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577-591 (1961).

Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931-954 (1958).

Les methodes directes en theorie des equations elliptiques. Prague: Editeurs Academia 1967.

Neumann, J. von [NU] Ober einen Hilfssatz der Variationsrechnung. Abh. Math. Sem. Univ. Hamburg 8, 28-31

(1931).

Nirenberg, L. [NI 1] On nonlinear elliptic partial differential equations and Holder continuity. Comm. Pure

Appl. Math. 6, 103-156 (1953). [NI 2] Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math. 8,

649-675 (1955). [NI 3] On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13, 115-162

(1959). [NI 4] Elementary remarks on surfaces with curvature of fixed sign. In: Proceedings of Symposia

in Pure Mathematics, Volume III, pp. 181-185. Providence, R.I.: American Mathematical Society 1961.

Nitsche, J. C. C. [NT] Vorlesungen tiber Minimalflachen. Berlin-Heidelberg-New York: Springer-Verlag 1975.

Novrusov, A. A. [NO] The modulus of continuity of the solution of the Dirichlet problem at a regular boundary

point. Mat. Zametki 12, 67-72 (1972) [Russian]. English translation in Math. Notes 12, no. I, 472-475 (1973).

Oddson. J. K. [OD] On the boundary point principle for elliptic equations in the plane. Bull. Amer. Math.

Soc. 74, 666--670 (1968).

Oleinik, 0. A. [OL] On properties of solutions of certain boundary problems for equations of elliptic type.

Mat. Sb. (N. S.) 30, 695-702 (1952) [Russian]. Oleinik, 0. A., and Radkevich, E. V. [OR] Second order equations with non-negative characteristic form. (Mathematical Analysis

1969). Moscow: Itogi Nauki 1971 [Russian]. English Translation: Providence, R.I., Amer. Math. Soc. 1973.

Oskolkov, A. P. [OS I] On the solution of a boundary value problem for linear elliptic equations in an unbounded

domain. Vestnik Leningrad. Univ. 16, no. 7, 38-50 (1961) [Russian]. [OS 2] A priori estimates of first derivatives of solutions of the Dirichlet problem for non uniformly

elliptic differential equations. Trudy Mat. Inst. Steklov 102, 105-127 (1967) [Russian]. English Translation in Proc. Steklov In st. Math. 102, 119-144 ( 1967).

Osserman, R. [OR I] On the Gauss curvature of minimal surfaces. Trans. Amer. Math. Soc. 96, 115-128 (1960). [OR 2] A Survey of Minimal Surfaces. New York: Van Nostrand 1969.

Pascali, D. [PA] Operatori Neliniari. Bucharest: Editura Academiei Republicii Socialiste Romania 1974.

Perron, 0. [PE] Eine neue Behandlung der Randwertaufgabe fiir Llu=O. Math. Z. 18, 42-54 (1923).

Piccinini, L. C. and S. Spagnola [PS] On the Holder continuity of solutions of second order elliptic equations in two variables.

Ann. Scuola Norm. Sup. Pisa (3) 26, 391-402 (1972).

Bibliography 393

Protter, M. H., and H. F. Weinberger [PW] Maximum Principles in Differential Equations. Englewood Cliffs, N.J. : Prentice-Hall 1967.

Pucci, C. [PU I]

[PU2] [PU 3]

Rad6, T. [RAJ

Rellich, R.

Su una limitazione per soluzioni di equazioni ellittiche. Boll. Un. Mat. Ita!. (3) 21, 228-233 (1966). Operatori ellittici estremanti. Ann. Mat. Pura Appl. (4) 72, 141-170 (1966). Limitazioni per soluzioni di equazioni ellittiche. Ann. Mat. Pura Appl. ( 4) 74, 15-30 ( 1966).

Geometrische Betrachtungen tiber zweidimensionale reguliire Variationsprobleme. Acta Litt. Sci. Univ. Szeged 2, 228-253 (1924-26).

[RE] Ein Satz tiber mittlere Konvergenz. Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl., 30-35 (1930).

Rodemich, E. [RO] The Sobolev inequalities with best possible constants. In: Analysis Seminar at California

Institute of Technology, 1966.

Royden, H. L. [RY] Real Analysis. 2nd ed., London: Macmillan 1970.

Schaefer, H. [SH] Uber die Methode der a priori Schranken. Math. Ann. 129, 415--416 (1955).

Schauder, J. [SCI] [SC 2]

[SC 3]

[SC 4]

[SC 5]

Serrin, J. [SE I]

[SE 2]

[SE 3]

[SE4]

[SE 5]

Der Fixpunktsatz in Funktionalraumen. Studia Math. 2, 17I-I80 (1930). Uberden Zusammenhang zwischen der Eindeutigkeit und Losbarkeit partieller Differential­gleichungen zweiter Ordnung vom elliptischen Typus. Math. Ann. 106, 661-721 (1932). Uber das Dirichletsche Problem im GroBen ftir nicht-lineare elliptische Differentialglei­chungen. Math. Z. 37, 623-634, 768 (1933). Uber lineare elliptische Differentialgleichungen zweiter Ordnung. Math. Z. 38, 257-282 (1934). Numerische Abschiitzungen in elliptischen linearen Differentialgleichungen. Studia Math. 5, 34--42 (1935).

On the Harnack inequality for linear elliptic equations. J. Analyse Math. 4, 292-308 (1955/56). Local behavior of solutions of quasi-linear elliptic equations. Acta Math. 111, 247-302 (1964). The problem of Dirichlet for quasilinear elliptic differential equations with many in­dependent variables. Philos. Trans. Roy. Soc. London Ser. A 264, 413--496 (1969). Gradient estimates for solutions of nonlinear elliptic and parabolic equations. In: Con­tributions to Nonlinear Functional Analysis, pp. 565-601. New York: Academic Press 1971. Nonlinear elliptic equations of second order. Lecture notes (unpublished), Symposium on Partial Differential Equations at Berkeley, 1971.

Simon, L. M. [SI I] Global estimates of Holder continuity for a class of divergence-form elliptic equations.

[SI 2]

[SI 3]

[SI 4] [SI 5]

[SI 6]

Simons, J.

Arch. Rational Mech. Anal. 56, 253-272 (1974). Boundary regularity for solutions of the non-parametric least area problem. Ann. of Math. 103, 429--455 (1976). Interior gradient bounds for non-uniformly elliptic equations. Indiana Univ. Math. J. 25, 821-855 (1976). Equations of mean curvature type in 2 independent variables. Pacific J. Math. (I977). Remarks on curvature estimates for minimal hypersurfaces. Duke Math. J. 43, 545-553 (1976). A Holder estimate for quasiconformal mappings between surfaces in Euclidean space, with application to graphs having quasiconformal Gauss map. Acta Math. (1977).

[SM] Minimal varieties in riemannian manifolds. Ann. of Math. (2) 88, 62-105 (1968).

394 Bibliography

Sobolev, S. L. [SO I] On a theorem of functional analysis. Mat. Sb. (N.S.) 4 (46), 471-497 (1938) [Russian].

English Translation in Amer. Math. Soc. Trans!. (2) 34, 39-Q8 (1963). [SO 2] Applications ofFunctional Analysis in Mathematical Physics. Leningrad: Izdat. Leningrad.

Spruck, J. [SP]

Gos. Univ. 1950 [Russian]. English Translation: Translations of Mathematical Mono­graphs, Vol. 7. Providence, R.I.: American Mathematical Society 1963.

Gauss curvature estimates for surfaces of constant mean curvature. Comm. Pure Appl. Math. 27, 547-557 (1974).

Stampacchia, G. [ST I] Contributi alia regolarizzazione delle soluzioni dei problemi al contorno per equazioni

del secondo orcline ellittiche. Ann. Scuola Norm. Sup. Pisa (3) 12, 223-245 (1958). [ST 2] I problemi a! contorno perle equazioni differenziali di tipo ellittico. In: Atti VI Congr.

Un. Mat. Ita!. (Naples, 1959), pp. 21-44. Rome: Edizioni Cremonese 1960. [ST 3] Problemi a! contorno ellittici, con dati discontinui, dotati di soluzioni holderiane. Ann.

Mat. Pura Appl. (4) 51, 1-37 (1960). [ST 4] Le probleme de Dirichlet pour Ies equations elliptiques du second ordre a coefficients

discontinues. Ann. Inst. Fourier (Grenoble) 15, fasc. I, 189-258 (1965). [ST 5] Equations elliptiques du second ordre a coefficients discontinues. Seminaire de Matbe­

matiques Superieures, No. 16 (Ete', 1965). Montreal, Que.: Les Presses de l'Universite de Montreall966.

Sternberg, S. [SB] Lectures on Differential Geometry. Englewood Cliffs, N.J.: Prentice-Hall 1964.

Talenti, G. [TA 1] Equazione lineari ellittiche in due variabili. Le Matematiche 21, 339-376 (1966). [TA 2] Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353-372 (1976).

Temam, R. [TE] Solutions generalisees de certaines equations du type hypersurface minima. Arch. Rational

Mech. Anal. 44, 121-156 (1971).

Trudinger, N. S. [TR 1] On Harnack type inequalities and their application to quasilinear elliptic equations.

[TR2]

[TR 3]

[TR4]

[TR5]

[TR6]

[TR 7]

[TR8] [TR 9]

Comm. Pure Appl. Math. 20, 721-747 (1967). On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473-483 (1967). Some existence theorems for quasi-linear, non-uniformly elliptic equations in divergence form. J. Math. Mech. 18, 909--919 (1968/69). On the regularity of generalized solutions of linear, non-uniformly elliptic equations. Arch. Rational Mech. Anal. 42, 50-Q2 (1971). The boundary gradient estimate for quasilinear elliptic and parabolic differential equations. Indiana Univ. Math. J. 21, 657-{)70 (1972). A new proof of the interior gradient bound for the minimal surface equation inn dimensions. Proc. Nat. Acad. Sci. U.S.A. 69, 821-823 (1972). Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa (3) 27, 265-308 (1973). Gradient estimates and mean curvature. Math. Z. 131, 165-175 (1973). A sharp inequality for subharmonic functions on two-dimensional manifolds. Math. Z. 133, 75-79 (1973).

[TR I 0] On the comparison principle for quasilinear divergence structure equations. Arch. Rational Mech. Anal. 57, 128-133 (1974).

[TR 11] Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients. Math. Z. (1977).

Ural'tseva, N. [UR] The solvability of the capillarity problem. Vestnik Leningrad. Univ. no. 19, 54-64 (1973)

[Russian].

Wahl, W. von [W A] Ober quasilineare elliptische Differentialgleichungen in der Ebene. Manuscripta Math. 8,

59-{)7 ( 1973).

Bibliography 395

Weinberger, H. F. [WE] Symmetrization in uniformly elliptic problems. In: Studies in Mathematical Analysis

and Related Topics, pp. 424-428. Stanford, Calif.: Stanford University Press 1962.

Widman, K.-0. [WI I] Inequalities for the Green function and boundary continuity of the gradient of solutions

of elliptic differential equations. Math. Scand. 21, 17-37 ( 1967). [WI 2] A quantitative form of the maximum principle for elliptic partial differential equations

with coefficients in L 00 • Comm. Pure Appl. Math. 21, 507-513 (1968). [WI 3] On the Holder continuity of solutions of elliptic partial differential equations in two

variables with coefficients in L 00 • Comm. Pure Appl. Math. 22, 669-682 (1969).

Williams, G. H. [WL] Existence of solutions for nonlinear obstacle problems. Math. z. 154, 51-65 (1977).

Winzell, B. [WN] Solutions of second order elliptic partial differential equations with prescribed directional

derivative on the boundary. Linkoping Studies in Science and Technology. Dissertations No. 003 (1975). Linkoping University, Sweden.

Yosida, K. [YO] Functional Analysis. 4th ed., Berlin-Heidelberg-New York: Springer-Verlag 1974.

Subject Index

adjoint 74, 79 formal- 172

apriori estimates 3 see gradient estimates, Holder estimates,

linear elliptic equations, maximum prin­ciples, quasilinear elliptic equations, Schauder estimates

area magnification factor 354

Banach space 68 barrier

for Laplace's equation 25 for linear equations 98 local- 25, 100 for non-uniformly elliptic equations 113 for quasilinear equations 278ff., 289 upper-(lower-) 98, 279

Bernstein's theorem for equations of mean curvature type 352 for minimal surface equation 378

bilinear form bounded- 78 coercive- 78

boundary manifold 225 boundary point lemma 33, 46 bounded linear mapping 69 bounded slope condition 225, 254, 259, 262 Brouwer fixed point theorem 238

Cauchy's inequality 138 co-area formula 356 compact imbedding of Banach spaces 160 compact mapping 70, 222

spectrum of- 73 comparison principles 207, 210, 292 completely continuous mapping

see compact mapping cone condition

exterior- 28, 193 uniform exterior- 195 uniform interior- 151

constant mean curvature equation Dirichlet problem, solvability 349 see also quasilinear elliptic equations

continuous imbedding 151 contraction mapping 69

difference quotient 1 04, 161 Dini continuity 66 Dirichlet integral 240, 358 Dirichlet problem

see constant mean curvature equation, Euler­Lagrange equation, Laplace's equation, linear elliptic equations, equations of mean curvature type, minimal surface equation, Poisson's equation, prescribed mean curvature equation, quasilinear el­liptic equations

distance function 381 divergence theorem 13 domain

with Ck· • boundary portion 89 of class Ck· • 88

dual space 74

eigenvalue 73 multiplicity 73

eigenvector 73 elliptic equations (operators),

see linear elliptic equations (operators), quasi-linear elliptic equations (operators)

elliptic parametric functional 375 extension of a function 130f. exterior boundary value problem 134 Euler-Lagrange equation 230

Dirichlet problem 23lff. global gradient estimate 309

see also quasilinear elliptic equations

Fredholm alternative 71, 79 fundamental solution 17

Gauss curvature 364 Gauss map 363

Holder estimates 369

Subject Index

generalized Dirichlet problem 4, 167 solvability 137 uniqueness 170 see also Dirichlet problem

generalized solution 3, 137, 166 see also weak solution

gradient estimates for equations of mean curvature type 371 for Euler-Lagrange equation 309 for harmonic functions 23, 29 for linear equations 48 for minimal surface equation 284, 286, 346 for Poisson's equation 40 for prescribed mean curvature equation 285,

287,290,308, 346f. for quasilinear equations

in divergence form 265, 268, 315, 320, 327 in general form 273, 276, 282, 283ff., 289ff.,

299, 308, 313, 327 in two variables 247, 268f.

Green's function for a ball 19 Dirichlet- 19 for oblique derivative problem 115

Green's identities 17 Green's operator 173 Green's representation formula 18

harmonic function 13 convergence theorems 2lf. derivative estimates 23, 29 on hypersurface 333 mean value property 14, 21 weakly- 28

harmonic lifting 24 Harnack's convergence theorem 22 Harnack inequality

for equations in two variables 40 for harmonic functions 16, 28 for linear equations 189 weak- 184, 200

Hessian matrix 381 Hilbert space 7 5 Holder coefficient 51 Holder conjugate 140 Holder continuity 51

De Giorgi estimate 190 for weak solutions 190

Holder estimates for equations in two variables 247ff. for Gauss map 369 for Newtonian potential 56 for Poisson's equation 59, 61, 117ff., 119 for quasiconformal maps 244f.

on hypersurface 362 for quasilinear equations 223, 229, 265, 268,

273, 276, 319 see also gradient estimates, Schauder estimates

Holder's inequality 138 generalized 139

Holder spaces 51 norms and seminorms 52, 60, 65, 84, 90

hypersurface 328

inner product 75 inner product space 75 interpolation inequalities

in Holder spaces 84, 89, 124ff., 136 in U spaces 139

John-Nirenberg inequality !59

Kellogg's theorem 65 Kelvin transform 66 Kondrachov compactness theorem 160

L" spaces 138 imbedding theorem !52 norm 138

Laplace-Beltrami operator 331 Laplace's equation I, 13ff.

Dirichlet problem solvability 20, 26 uniqueness 15

fundamental solution 17 maximum principle 15 see also harmonic functions

Laplacian 13, 331 Lax-Milgram theorem 78

397

Leray-Schauder fixed point theorem 222, 228 linear elliptic equations (operators) I, 30f., 82 ff.,

!66ff. classical solutions

boundary regularity 105, 133 Dirichlet problem

Fredholm alternative 102 solvability 95ff., 101, 109f., 112,135,177 uniqueness 32, 36

global bound 35 global regularity 106 interior regularity 103f. maximum principles 3lff., 136 Neumann problem 34 oblique derivative problem 120ff. Schauder estimates 85, 88, 90ff., 107, 134f.

in divergence form 3, 45f., 166ff. see below weak solutions

in two variables Holder estimates 247f.

weak solutions 3, 166, 178 boundary estimates 192ff. Dirichlet problem

Fredholm alternative 172 solvability 170, 196 uniqueness 170

398

linear elliptic equations-contd. weak solutions-contd.

global bound 179, 181 global regularity 176f. Holder continuity 190, 192 interior regularity 173ff. local bound 184 maximum principles 45, 168, 188, 199

linear mapping, bounded 69 Lipschitz continuity 51 Liouville's theorem 29, 43

MP spaces 157 maximum principles

for Laplace's equation 15 for linear equations

classical solutions 3lff., 136 weak solutions 45, 168, 188, 199

for quasilinear equations 206, 208, 213ff., 219f.

gradient- 303 mean curvature 330, 381, 384

generalized- 287 mean curvature type, equations of 350, 369ff.

curvature estimate 370 gradient estimate 371

mean value inequality 14, 28, 67 on hypersurface 333, 336, 379

mean value theorem 14 method of continuity 70 minimal surface 384 minimal surface equation (operator)

curvature estimate 377 Dirichlet problem 297, 347 gradient estimates 284, 286, 346 see also quasilinear elliptic equations (operators)

minimal surface type, equations of 3 77 see also mean curvature type, equations of

mixed boundary condition 47 mixed boundary value problem 198 mollifier 140 Morrey's lemma 243 Moser iteration technique 180, 185

Newtonian potential 18, 50ff. differentiability properties 53 f. Holder estimates 56, 63

non-uniformly elliptic equations (operators) Iliff.

see also linear elliptic equations (operators) norm 68

boundary- 89 on HOlder spaces 52, 60, 65, 84, 90 on LP spaces 138

normed linear space 68

oblique derivative problem 115ff. for linear equations

Schauder estimates 120f.

solvability 122 non-regular- 134 for Poisson's equation 115ff.

Holder estimates 117, 119 regular- 115

Subject Index

oblique derivative boundary condition 115, 120 regular- 47, 115, 120

orthogonality 7 6 orthogonal projection 77

parallelogram law 75 parametric functional 375

elliptic- 375 partition of unity 130 perpendicular elements 76 Perron method 24, 97 Perron solution 25 Poincare inequalities 157 Poisson's equation 50ff.

boundary Holder estimates 63ff. compactness of solutions 60 Dirichlet problem 15, 55, 61, 66, 67 gradient estimates 40 interior Holder estimates 59, 61 oblique derivative problem 115ff.

Poisson integral 20 Poisson integral formula 20 Poisson kernel 20 prescribed mean curvature equation 205, 328ff.

boundary gradient estimate 285, 290 gradient estimate 308 maximum principle 218f. Dirichlet problem

non-solvability 297 solvability 347, 380 uniqueness 347

see also quasilinear elliptic equations principal coordinate system 381 principal curvature 381 principal directions 381

quasiconformal mapping 239ff. Holder estimates 244 on hypersurface 356

Holder estimates 362 quasilinear elliptic equations (operators) 1,

201 ff. comparison principles 207, 210, 292 in divergence form 5, 204

Dirichlet problem, solvability 225, 276, 326 global gradient estimates 268, 315, 320 interior gradient estimates 265, 320, 327 regularity 319

equivalent- 203 in general form

boundary gradient estimates 282, 283ff., 289ff., 299

continuity estimate 298

Subject Index

Dirichlet problem non-solvability 295, 297 non-uniqueness 209 solvability 223, 229, 276, 32lff., 326 uniqueness 208

global gradient estimates 276, 308 interior gradient estimates 273, 313 regularity 223, 229

maximum principles 206, 208, 213ff., 219ff. for gradient 303

in two variables Dirichlet problem, solvability 225, 250,

257,276 gradient estimates 256, 268f.

quasisolution 233

reflexive space 74 regular boundary point 25, 133, 196 regular functional 231 regular oblique derivative problem

see oblique derivative problem regularity

of classical solutions l03ff., 133 of weak solutions l73ff. for quasilinear equations 223, 229, 319 see also gradient estimates, Holder estimates,

Schauder estimates regularization of a function 140 resolvent operator 73 Riesz representation theorem 77

scalar product 75 Schauder estimates

boundary 90ff., 94 global 93, 135 interior 85, 88, Hi7, 134 for oblique derivative problem l20f.

Schauder fixed point theorem 221 Schwarz inequality 75, 139 Schwarz reflection principle 28 segment condition 148 Sobolev inequalities 148, 151, 157, 379

best constant 151 Sobolev spaces 146, l83ff.

density theorems 147 imbedding theorems 148, 151, 155, 160 norm 146 scalar product 147

sphere condition enclosing- 284 exterior- 27 interior- 32

Stokes' theorem 353

straightening the boundary 89 strict exterior plane condition 112

399

strictly elliptic equations (operators) 30, 82 see also linear elliptic equations (operators),

quasilinear elliptic equations (operators) strong derivative 143 strong maximum principle

see maximum principles structure conditions 178, 184, 213, 250, 280ff.,

307ff. natural- 308, 315

structural inequalities see structure conditions

subfunction 24,97 subharmonic function 13, 23

on hypersurface 333 weakly- 28

subsolution 32, 45, 97 weak- 178

superfunction 24, 97 superharmonic function 13, 23

weakly- 28 supersolution 32, 45, 97

weak- 178

tangential gradient 329 test function 16 7 three-point condition 255, 259 total variation 164 triangle inequality 75

uniformly convex domain 225, 284 uniformly elliptic equations (operators) l, 30,

203 see also linear elliptic equations (operators),

quasilinear elliptic equations (operators)

variational problems 230ff.

weak convergence 80 weak derivative 142

chain rule 144 product rule 143

weak maximum principle see maximum principles

weak solution 3, 166 see also linear elliptic equations

Wirtinger's inequality 242

Young's inequality 138

Notation Index

Spaces and their norms and seminorms ll·[[wk. P(Q) 146

BV(Q) 165 Other notation (see also page 8) C\Q), Ck(Q), C~(.Q) 9 ca(Q), ca(Q), Ck·a(Q), Ck·a(Q) 51 A(6R) 333 Ck·a(oQ), Ck·a(T)(TcoQ) 89 a~, a:!, 287 C!•a(Q) 81, 127 a~ 300,305,370 Hk(Q), H~(Q) 147 b0 , boo 287 U(Q) 138 b* 370 Lfoc(Q) 140 C; 300, 305 MP(Q) 157 Tl (r, z), Tl (r) 240 Wk(Q) 143 Tl (p, Z) 358 Wk·P(Q) 146 L1 13, 331 W~·P(Q) 147 L1h, L1? 104 w~~/(Q) 147 dx, dx,y 60 l·lk; ,Q ' l·lk, a; ,Q 52 ax, ax,y 65 1·1~; ,Q ' 1·1~, a; ,Q 52 d(x) 381 l·lt; a, I·IL; a 60 b 302, 329 l·lt, !JuT' l·lt a; .QuT 65, 90 bk 264 l·ltb, l·lk~~;Q 84 () 304

1·1~! a; ,Q 60 b;b; 331 l·l~!a;!JuT 65 0; 310 [· ]a;xo 51 Iff 203 [·Ja;D 51 Iff* 292 [ · ]k;.Q' [ · Jk,a;.Q 52 e1 104 [·Jt;a, [·Jt,a;!J 60 ff 279 [·JtauT,[·Jt,a;!JuT 65,90 [ ·](G) [·](G) 84

k;!J' k,o:;,Q

G(x, y) 19, 115 G 363

Hlck(n)' Hlck."<i1l 52 11·11 ~k(Q)' 11·11 ~k, "(Ql 52 11·11 68, 69

r(x-y) 17 H 341, 381 J(q>) 354

HIBV(.Q) 165 .%~ .%- 287

Hlp, HILP(.Q) 138 HIMP(,Q) 157

k, k(R) 184 k, k(R) 190

Notation Index

K; 381 E(u, v) 166 m 284 v 13, 381 6R, 6R(y) 334 f7 284

ff * u+,u-

302

uif, uM uh 140 v,.. 152

32, 145 193

(oQ, cp) 225

401

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Eine Auswahl

23. Pasch: Vorlesungen iiber neuere Geometrie 41. Steinitz: Vorlesungen iiber die Theorie der Polyeder 45. Alexandroff/Hopf: Topologie. Band I 46. Nevanlinna: Eindeutige analytische Funktionen 63. Eichler: Quadratische Forrnen und orthogonale Gruppen

I 02. NevanlinnajNevanlinna: Absolute Analysis 114. Mac Lane: Homology 123. Yosida: Functional Analysis 127. Hermes: Enumerability, Decidability, Computability 13 I. Hirzebruch: Topological Methods in Algebraic Geometry 135. Handbook for Automatic Computation. Vol. !/Part a: Rutishauser: Description of ALGOL 60 136. Greub: Multilinear Algebra 137. Handbook for Automatic Computation. Vol. I /Part b: Grau/Hill/Langmaack:

Translation of ALGoL 60 138. Hahn: Stability of Motion 139. Mathematische Hilfsmittel des Ingenieurs. I. Teil 140. Mathematische Hilfsmittel des Ingenieurs. 2. Teil 141. Mathematische Hilfsmittel des Ingenieurs. 3. Teil 142. Mathematische Hilfsmittel des Ingenieurs. 4. Teil 143. Schur/Grunsky: Vorlesungen iiber Invariantentheorie 144. Weil: Basic Number Theory 145. Butzer/Berens: Semi-Groups of Operators and Approximation 146. Treves: Locally Convex Spaces and Linear Partial Differential Equations 147. Lamotke: Semisimpliziale algebraische Topologie 148. Chandrasekharan: Introduction to Analytic Number Theory 149. Sario/Oikawa: Capacity Functions 150. IosifescujTheodorescu: Random Processes and Learning 151. Mandl: Analytical Treatment of One-dimensional Markov Processes 152. Hewitt/Ross: Abstract Harmonic Analysis. Vol. 2: Structure and Analysis for

Compact Groups. Analysis on Locally Compact Abelian Groups 153. Federer: Geometric Measure Theory 154. Singer: Bases in Banach Spaces I 155. Miiller: Foundations of the Mathematical Theory of Electromagnetic Waves 156. van der Waerden: Mathematical Statistics 157. ProhorovjRozanov: Probability Theory. Basic Concepts. Limit Theorems. Random Processes 158. Constantinescu/ Cornea: Potential Theory on Harmonic Spaces 159. Kothe: Topological Vector Spaces I 160. AgrestjMaksimov: Theory of Incomplete Cylindrical Functions and their Applications 161. BhatiajSzegii: Stability Theory of Dynamical Systems 162. Nevanlinna: Analytic Functions 163. Stoer/Witzgall: Convexity and Optimization in Finite Dimensions I 164. SariojNakai: Classification Theory of Riemann Surfaces 165. MitrinovicjVasic: Analytic Inequalities 166. Grothendieck/Dieudonne: Elements de Geometrie Algebrique I 167. Chandrasekharan: Arithmetical Functions 168. Palamodov: Linear Differential Operators with Constant Coefficients 169. Rademacher: Topics in Analytic Number Theory 170. Lions: Optimal Control of Systems Govenied by Partial Differential Equations 171. Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces 172. Biihlmann: Mathematical Methods in Risk Theory 173. Maeda/Maeda: Theory of Symmetric Lattices 174. Stiefel/Scheifele: Linear and Regular Celestial Mechanics. Perturbed Two-body

Motion-Numerical Methods-Canonical Theory

175. Larsen: An Introduction to the Theory of Multipliers 176. Grauert/Remmert: Analytische Stellenalgebren 177. Fliigge: Practical Quantum Mechanics I 178. Fliigge: Practical Quantum Mechanics II 179. Giraud: Cohomologie non abelienne 180. Landkof: Foundations of Modern Potential Theory 181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications I 182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications II 183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications III 184. Rosenblatt: Markov Processes. Structure and Asymptotic Behavior 185. Rubinowicz: Sommerfeldsche Polynommethode 186. Handbook for Automatic Computation. Vol. 2. Wilkinson/Reinsch: Linear Algebra 187. Siegel/Moser: Lectures on Celestial Mechanics 188. Warner: Harmonic Analysis on Semi-Simple Lie Groups I 189. Warner: Harmonic Analysis on Semi-Simple Lie Groups II 190. Faith: Algebra: Rings, Modules, and Categories I 192. Mal'cev: Algebraic Systems 193. P6lya/Szego: Problems and Theorems in Analysis I 194. lgusa: Theta Functions 195. Berberian: Baer*-Rings 196. Athreya/Ney: Branching Processes 197. Benz: Vorlesungen iiber Geometrie der Algebren 198. Gaal: Linear Analysis and Representation Theory 199. Nitsche: Vorlesungen iiber Minimalfliichen 200. Dold: Lectures on Algebraic Topology 201. Beck: Continuous Flows in the Plane 202. Schmetterer: Introduction to Mathematical Statistics 203. Schoeneberg: Elliptic Modular Functions 204. Popov: Hyperstability of Control Systems 205. Nikol'skii: Approximation of Functions of Several Variables and Imbedding Theorems 206. Andre: Homologie des Algebres Commutatives 207. Donoghue: Monotone Matrix Functions and Analytic Continuation 208. Lacey: The Isometric Theory of Classical Banach Spaces 209. Ringel: Map Color Theorem 210. Gihman/Skorohod: The Theory of Stochastic Processes I 211. Comfort/Negrepontis: The Theory of Ultrafilters 212. Switzer: Algebraic Topology-Homotopy and Homology 213. Shafarevich: Basic Algebraic Geometry 214. van der Waerden: Group Theory and Quantum Mechanics 215. Schaefer: Banach Lattices and Positive Operators 216. P6lya/Szego: Problems and Theorems in Analysis II 217. Stenstrom: Rings of Quotients 218. Gihman/Skorohod: The Theory of Stochastic Processes II 219. Duvaut/Lions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry 1: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. Bergh/LOfstrom: Interpolation Spaces. An Introduction 224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order 225. Schutte: ProofTheory