17

REGULARITY AT THE BOUNDARY AND REMOVABLE SINGULARITIES FOR

SOLUTIONS OF QUASILINEAR PARABOLIC EQUATIONS

William P. Ziemer

1. INTRODUCTION

The purpose of this note is to describe recent results concerning

removable singularities and behavior of weak solutions of quasilinear

parabolic equations of the second order at the boundary of an arbitrary

domain. Specifically, .7e investigate the local behaviour of weak solutions

of equations of 'che form

(1)

where A and B are, respectively, vector and scalar valued Borel

functions defined on "x Rl x Rn , where " is arbitrary open subset

The functions A and B are required to satisfy ·the

following. structure conditions:

Here, p > 1, 0.0 > 0 b O > 0 and the remaining coefficients are non-

negative functions of (x,t) that are required to belong to specified

Lebesgue classes. For the purposes of this exposition, "e will simply

require a P 1

, aP 2

and to lie in LqW) where

n 1 1 + - <

pq q

18

Essentially all research concerned with the behavior of weak solutions

of (1) has been ·restricted to the case p = 2 • For example, interior

regularity, i.e., Holder continuity of weak solutions of (1) ha.s been

established by several authors, c.f., [KO], [LSU], [AS], [T]. Landis,

[L], apnounced a criterion for continuity of solutions of the heat equation

at the boundary of an arbitrary open set, although a complete development of

his results has apparently never appeared. Recently, Evans and Gariepy,

[EG], established a characterization of regular boundary points for the

heat equation which is in the same spirit as the Wiener criterion for

Laplace's equation. Other results concerning boundary regularity of

linear parabolic equations inclune [E], [Ll], [L2], [PI], [EK].

Closely associated with the problem of regularity at the boundary of

weak solutions of (1) is the question of determining conditions under which

a compact set K c Q is removable for solutions of (1). Some results in

this direction were obtained in [A] for linear equations and in [EP] for

equations of the form (1) with p = 2 For a general development of

removability results for a wide class of higher order linear equations,

the reader is referred to [HP].

2. CONTINUITY AT THE BOUNDARY

If U c Q is an open set, a bounded function is said

to be a weak solution of (1) in U if

for all $ E C~(U)

The fundamental solution of the heat operator H is

given by

19

t > 0

G{x,t)

t :$; 0 .

For any set E C Rn+l, the alassiaal parabolia aapaaity is qefined by

where the supremum is taken over all non-negative Borel measures ~

supported in E whose potential, G * ~, is everywhere bounded above

by 1 •

For the purpose of describing boundary regularity results, let

R (r) a

For a > 0, let

where B{xO,r) is the ball in Rn with center xo and radius r.

We associate with Ra{r) a subcylinder

* r 2 2 1 2 Ra{r) = B{XO'2) x (to - ~r ,to ~r)

U € Wl ,2{Q) , we say that

u{zo) :$; k weakly

if for every t > k , there is an r > 0 such that

whenever n € C~[u{zo,r)] where u{zO,r) denotes the ball in Rn+l with

center Zo and radius r A similar definition is given for u{zo) ~ k

weakly and consequently it is clear what is meant by u{zO) = k weakly.

The following result comes from [Zl] and [GZ2].

THEOREl"! Let Q ----U E Wl ,2(Q) be a

p 2 such that

(3)

then

20

be an open subset of Rn+l with

bouniled weak solution of (1) with

u(zo) = k weakly. If for some

lim u(z) k.

z+zO

ZEQ

'" ,

Zo E aQ • Let

structure (2) and

C/, > 0 ,

In the case that Q = D x (O,T) where D is an open subset of

Rn , it is no·t difficult to show that if Zo = (XO/tO) E aQ, then (3)

holds if and only if

(4)

where is classical Newtonian capacity in Rn But (4) is precisely

the Wiener criterion for continuity of solutions of Laplace's equation at

Hence, we have ·the following.

COROLLARY If Xo E 3D is a regular point for Laplace's equation, then

Zo = (Xo,tO) E 3[Dx(0,T)] is a regular point for bouniled weak solutions

of (1) with structure (2) anil p 2.

The author has recently proved that weak bounded solutions of (1), (2)

with P > 1 are continuous in Q, [Z2j . However, the question of

extending the above theorem to cover the case of all p > 1 remains open.

21

3. REMOVABLE SINGULARITIES

Suppose D C Rn is an open set and KeD a compact set of zero

Newtonian capacity. It is well known that a bounded harmonic function

defined in D - K can be defined on K in such a way that the resulting

function is harmonic throughout D Likewise a single point is removable

for any harmonic function which is o(r2- n ) in a neighbourhood of the

point. These two results represent extreme cases concerning the size of

the singular set. Carleson [C] provided an interpolation between them by

relating the Hausdorff dimension of the singular set to the Lebesgue class

of the harmonic function. Serrin [S] extended Carleson's results to

solutions of the elliptic counterpart of (1) and (2) by using the notion

of s-capacity in place of Hausdorff measure. Below we state analogous

results for solutions of parabolic equations of the form (1).

One of the difficulties encountered in the parabolic situation is

the selection of the appropriate capacity which is used to balance the

size of the singular set against the Lebesgue class of the solution. Unlike

the corresponding elliptic case, no obvious definition is suggested by the

analysis of the equation.

compact set K as

r (K) s

The capacity we employ is defined for each

where the infimum is taken over all v E C~(Rn+l) such that v ~ Ion.

K • Here, s > 1 1 1 1 and 1I~I_l,S' , -+ 8'= s denotes the norm of

dV when taken element of wI,s' (Rn ) dt(· ,t) as an It can be shown that,

for each compact set K c Rn+l, r 2 (K) = C(K) where C is classical

parabolic capacity defined in §2 above, c.f., [PM]. Also, it is shown in

22

[GZ2] t.hat r s

is strictly 1Meaker than the capacity employed in [Al or

[EP] in the sense that there exist, compact sets K whose capacity, as

defined in [Al or [EP] , is positive but r (K) s o If (K) ~ C(K) = 0 ,

it can be shm'ln [GZlj that there is a sequence of smooth functions {cc} l

with the following properties

Os()"sl l

()'i 0 on a neighbourhood of K

at. + 0 l

a.e, as

where is the adjoint to the heat operator.

is critical in establishing the following result" [GZll.

This information

THEOREM Let K be a compact subset of an open set Q C Rn +l , Let

U E L~ (n) n l,2W .J be a weak soZution in Q - K of (1) and (2) lU1:th oc W10c ' ,-K

P 2 If r 2 (K) o s then u E w1 ,2 (rn 10c

and U is a weak soZution of

(1) in Q

Clearly this result is optimal for the class of equations under

considera'tion, for if r 2 (K) > 0, then ,:he capacity equilibrium potential

of K is a bounded function on Rn+1 that satisfies the hea't equation on

Rn+l _ K but not on

In order to consider weak solutions of (1), (2) for all p > 1 ,

A typical we assume that the constant b O that appears in (2) is zero.

interpolatory result analogous to that of s.errin"s cited above takes the

following form, [Z2j.

23

Let K c Q be a compact setl;)ith r s (K) ~ 0 0 If'

LC[ (Q', n li'P W- K) lac lac'

with P :S S is a "leak soluMon of (I] f (2)

-then u -is a 'UJeak sotution in Q l?l"Jovided (p-l) (_5_, < q ~ 's-p' -

A more de·tailed ax!alysis caT! be provid,ad by co.{lsidering sol'v;tions

t.11d.'C lie i:~L the spa.ces

( ,

U lui Clip 'jl/q

c1tr ~

REFERENCES

rfBelected pro.ble:c[~s on exceptional se ts:' 1 r

hefrt equ.a:ticn" y to appear ~

[EPl

Soc, 2 (1970), 273-283.

Effros, E~G" f a.nd Kasdarl. p :J~IJ<>! elOn the Dirichlet: p~coblem for

[El Eklund, 1\1" f 11Boundary behav"ior of solu-tions of parabolic equations

788-792.

[GZl]

[GZ2]

24

Gariepy, R., and Ziemer, w., "Removable sets for quasilinear

parabolic equations", J. London Math. Soa •• 21 (1980),311-318.

Gariepy, R., and Ziemer, W., "Thermal capacity and boundary

regularity", to appear.

[HP] Harvey, R., and Pol king , J., "A notion of capaci tywhich

characterizes removable singularities", Trans. Amer. Math. Soa.

169 (1972),183-195.

[KD] Kruzkov, S., and Dleinik, D., "Quasilinear second order parabolic

equations with many independent variables", Russian Mathematiaal

Surveys. 16 (1961),106-146.

[LSU] Ladyzenskaja, D.A., Solonnikov, V.A., and Ural'ceva, N.N.,

Linear and Quasilinear Equations of Parabolia Type. Nauka, Moscow

1967, English translation, Amer. Math. Soc., Translations of

Mathematical Monographs, Vol. 23, 1968.

[Ll] Lanconelli, E., "SuI problema di Dirichlet per l' equazione del

calore", Ann. di Mat. Pura ed Appl; ~ (1973), 83-114.

[L2] Lanconelli, E., "SuI problema di Dirichlet per equazione

paraboliche del secondo ordine a coefficiente discontinui", Ann. di

Math. Pura ed Appl. 106 (1975),11-38.

[L] Landis, E.M., "Necessary and sufficient conditions for the

regularity of a boundary point for the Dirichlet problem for the

heat equation", Soviet Math. DoH. 10 (1969), 380-384.

[PEl Petrowski, I., "Zur ersten Randwertaufgabe der Warmeleitungsgleichung",

Composito Math. 1 (1935),383-419.

[PI] Pini, B., "Sulla regolarita e irregolarita della frontiera per il

prima problema di valori al contorns relativo all'equazione del

calore", Ann. di Mat. Pura ed Appl. 40 (1955), 69-88.

[PM] Pierre, M., "Parabolic capacity and Sobolev spaces", MRC Teah.

Report. #2109 (1980).

[S] Serrin, J., "Local behavior of solutions of quasilinear

equations", Aata Math .• 111 (1964), 247-302.

25

[Tl Trudinger, N., "Pointwise estimates and quasilinear para.bolic

equations", Corrun. Pure Appl. Math. 21 (1968), 205-266.

[Zl] Ziemer I W.P ,_ , "Behavior a-t the boundary of solutions of

quasilinear parabolic equations", J. Diff. Eqnso ~ (1980),

291-305.

[Z2] Ziemer, W.P., "Local behavior of quasilinear parabolic equations

with non-linear growth", to appear.

Department of Mat_hematics, Indiana Universi-ty I Bloomington, INDIANA 47401 U.S.A.