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Besov regularity of solutions of the p-Laplace equation
Benjamin Scharf
Technische Universitat Munchen,Department of Mathematics,Applied Numerical Analysis
benjamin.scharf@ma.tum.de
joint work with Lars Diening (Munich),Stephan Dahlke, Christoph Hartmann, Markus Weimar (Marburg)
Jena, June 27, 2014
Overview
Introduction and results for the Laplace equation (p = 2)Introduction to the p-LaplaceApproximation in Sobolev and Besov spacesKnown results for the Laplace equation (p = 2)
Sobolev and local Holder regularity of the p-LaplaceSobolev regularity of the p-LaplaceLocal Holder regularity of the p-Laplace equation
Besov regularity of solutions of the p-Laplace equationFrom Bs
p,p(Ω) and C `,αγ,loc(Ω) to Bσ
τ,τ (Ω)Besov regularity of the p-Laplace
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 2 of 23
Introduction and known results – Introduction to the p-Laplace
The p-Laplace - Introduction
Ω ⊂ Rd Lipschitz domain, d dimension, 1 < p <∞
Inhomogeneous p-Laplace equation:
∆pu := div(|∇u|p−2∇u
)= f in Ω, u = 0 on ∂Ω.
Variational (weak) formulation:∫Ω
⟨|∇u|p−2∇u,∇v
⟩dx =
∫Ω
f v dx for all v ∈ C∞0 (Ω)
has a unique solution u ∈ W 1p (Ω) for f ∈W−1
p′ (Ω),
has model character for nonlinear problems, similar to the Laplaceequation (p = 2) for linear problems
nice and free introduction: P. Lindqvist. Notes on the p-Laplace equation, 2006.
http: // www. math. ntnu. no/ ~ lqvist/ p-laplace. pdf
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 3 of 23
Introduction and known results – Approximation in Sobolev and Besov spaces
Sobolev and Besov spaces
W sp (Ω): Sobolev space of smoothness s and integrability p on Ω
Bsp,p(Ω): Besov space of smoothness s and integrability p on Ω
Wavelet representation: ηI ,p = |I |1/2−1/p ηI p-normalized wavelets
g ∈ Bsp,p(Rd )⇔ g = P0(g) +
∑I
∑η∈Ψ
⟨g , ηI ,p′
⟩ηI ,p
and∥∥∥P0(g) Lp(Rd )
∥∥∥+∥∥∥⟨g , ηI ,p′
⟩bs
p,p(Rd )∥∥∥ <∞
Here ∥∥∥⟨g , ηI ,p′⟩
bsp,p(Rd )
∥∥∥p=∑
I
∑η∈Ψ
|I |−sp/d∣∣⟨g , ηI ,p′
⟩∣∣pmore smoothness ⇔ more decay of the wavelet coefficients
Trivial embedding: Bs+εp,p (Ω) →W s
p (Ω) → Bsp,p(Ω)
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 4 of 23
Introduction and known results – Approximation in Sobolev and Besov spaces
Sobolev and Besov spaces
W sp (Ω): Sobolev space of smoothness s and integrability p on Ω
Bsp,p(Ω): Besov space of smoothness s and integrability p on Ω
Wavelet representation: ηI ,p = |I |1/2−1/p ηI p-normalized wavelets
g ∈ Bsp,p(Rd )⇔ g = P0(g) +
∑I
∑η∈Ψ
⟨g , ηI ,p′
⟩ηI ,p
and∥∥∥P0(g) Lp(Rd )
∥∥∥+∥∥∥⟨g , ηI ,p′
⟩bs
p,p(Rd )∥∥∥ <∞
Here ∥∥∥⟨g , ηI ,p′⟩
bsp,p(Rd )
∥∥∥p=∑
I
∑η∈Ψ
|I |−sp/d∣∣⟨g , ηI ,p′
⟩∣∣pmore smoothness ⇔ more decay of the wavelet coefficients
Trivial embedding: Bs+εp,p (Ω) →W s
p (Ω) → Bsp,p(Ω)
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 4 of 23
Introduction and known results – Approximation in Sobolev and Besov spaces
Linear and Adaptive approximation by wavelets (i)
How to approximate f ∈ Bsp,p(Ω), Ω bounded, by wavelet basis?
Linear approximation fk of f (order k : ∼ 2kd terms):
fk = P0(g) +∑|I |≥2−k
∑η∈Ψ
⟨g , ηI ,p′
⟩ηI ,p
It holds
f ∈ Bsp,p(Ω) (or W s
p (Ω))⇒ ‖f − fk Lp(Ω)‖ . 2−ks .
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 5 of 23
Introduction and known results – Approximation in Sobolev and Besov spaces
Linear and Adaptive approximation by wavelets (ii)
Adaptive approximation fk of f (order k : ∼ 2kd terms):
f Dk = P0(g) +
∑(I ,η)∈D
⟨g , ηI ,p′
⟩ηI ,p with |D| = 2kd
best m-term approximation: choose D to minimize∥∥∥f − f Dk Lp(Ω)
∥∥∥ : take 2kd largest wavelet coefficients!
Let 1τ = σ
d + 1p , in particular τ < 1 possible. It holds
f ∈ Bστ,τ (Ω)⇒ ‖f − fk Lp(Ω)‖ ∼ 2−kσ
Besov regularity is the maximal possible convergence rate of anadaptive algorithm ⇒ how much higher than Sobolev regularity?
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 6 of 23
Introduction and known results – Approximation in Sobolev and Besov spaces
Linear and Adaptive approximation by wavelets (iii)
The main reason is the following computation:
TheoremLet 1
τ = σd + 1
p , x ∈ `τ and x∗ its non-increasing rearrangement. Then
‖x∗ − x∗k‖p ≤ k−σd ‖x‖τ ,
where x∗k is the cut-off of x∗ after the k first terms.
Proof:Assume w.l.o.g. that ‖x‖τ = 1. Then
|x∗(j)|τ ≤ |x∗(k)|τ ≤ 1
k‖x∗‖ττ =
1
k· for j > k .
Therefore
‖x∗ − x∗k‖pp ≤ ‖x∗ − x∗k‖p−τ
∞ · ‖x∗ − x∗k‖ττ ≤ kτ−pτ · 1 = k−
σd
p.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 7 of 23
Introduction and known results – Known results for the Laplace equation (p = 2)
Sobolev regularity for p = 2, the linear case
Theorem (Jerison, Kenig 1981,1995, Theorem B)
Positive: Lipschitz domain Ω ∈ Rd , f ∈ L2(Ω). Then the solution u of
∆u = f in Ω, u = 0 on ∂Ω
belongs to W3/22 (Ω).
Negative: For any s > 3/2 there exists a Lipschitz domain Ω andsmooth f s.t. u with
∆u = f in Ω, u = 0 on ∂Ω
does not belong to W s2 (Ω).
Careful! ∃ C1-domain Ω and f ∈W−1/22 (Ω) such that u /∈W
3/22 (Ω)
D. Jerison, C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains. J.Funct. Anal. 130, 161–219, 1995.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 8 of 23
Introduction and known results – Known results for the Laplace equation (p = 2)
Besov regularity for p = 2 (i)
Theorem (Dahlke,DeVore ’97; Jerison,Kenig ’95; Hansen 2013)
Lipschitz domain Ω ∈ Rd , f ∈W γ2 (Ω) for γ ≥ max
(4−d
2d−2 , 0)
. Then
the solution u of
∆u = f in Ω, u = 0 on ∂Ω
belongs to Bστ,τ (Ω), 1
τ = σd + 1
p , for any σ < 32 ·
dd−1 .
Besov reg. always better than 3/2, the maximal Sobolev regularity proof by a general embedding:
small global Sobolev regularity + better local (weighted) Sobolevregularity (Babuska-Kondratiev) result in better Besov regularity!
S. Dahlke, R.A. DeVore. Besov regularity for elliptic boundary value problems. Comm.Partial Differential Equations, 22(1–2), 1–16, 1997.
M. Hansen, n-term approximation rates and Besov regularity for elliptic PDEs onpolyhedral domains, to appear in J. Found. Comp. Math.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 9 of 23
Introduction and known results – Known results for the Laplace equation (p = 2)
Besov regularity for p = 2 (i)
Theorem (Dahlke,DeVore ’97; Jerison,Kenig ’95; Hansen 2013)
Lipschitz domain Ω ∈ Rd , f ∈W γ2 (Ω) for γ ≥ max
(4−d
2d−2 , 0)
. Then
the solution u of
∆u = f in Ω, u = 0 on ∂Ω
belongs to Bστ,τ (Ω), 1
τ = σd + 1
p , for any σ < 32 ·
dd−1 .
Besov reg. always better than 3/2, the maximal Sobolev regularity proof by a general embedding:
small global Sobolev regularity + better local (weighted) Sobolevregularity (Babuska-Kondratiev) result in better Besov regularity!
S. Dahlke, R.A. DeVore. Besov regularity for elliptic boundary value problems. Comm.Partial Differential Equations, 22(1–2), 1–16, 1997.
M. Hansen, n-term approximation rates and Besov regularity for elliptic PDEs onpolyhedral domains, to appear in J. Found. Comp. Math.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 9 of 23
Introduction and known results – Known results for the Laplace equation (p = 2)
Besov regularity for p = 2 (ii)
Proof Idea:
extend u to Rn and take its wavelet decomposition – 3 parts
1. father wavelets (independent of regularity)2. interior and exterior wavelets ηI ,p with
dist(I , ∂Ω) & diam(I ) (1)
3. boundary wavelets ηI ,p; (1) doesn’t hold
handle 3 parts separately
1. no problem2. use weighted Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W 2
2 (Ω,w),weigth w exploding at the boundary (Babuska-Kondratiev spaces)
3. use global Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W3/22 (Ω), use
counting argument:
#ηI ,p boundary wav.,diam(I ) ∼ 2−j ∼ 2j(d−1)
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 10 of 23
Introduction and known results – Known results for the Laplace equation (p = 2)
Besov regularity for p = 2 (ii)
Proof Idea:
extend u to Rn and take its wavelet decomposition – 3 parts
1. father wavelets (independent of regularity)2. interior and exterior wavelets ηI ,p with
dist(I , ∂Ω) & diam(I ) (1)
3. boundary wavelets ηI ,p; (1) doesn’t hold
handle 3 parts separately
1. no problem2. use weighted Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W 2
2 (Ω,w),weigth w exploding at the boundary (Babuska-Kondratiev spaces)
3. use global Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W3/22 (Ω), use
counting argument:
#ηI ,p boundary wav.,diam(I ) ∼ 2−j ∼ 2j(d−1)
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 10 of 23
Sobolev and local Holder regularity
Table of contents
Introduction and results for the Laplace equation (p = 2)Introduction to the p-LaplaceApproximation in Sobolev and Besov spacesKnown results for the Laplace equation (p = 2)
Sobolev and local Holder regularity of the p-LaplaceSobolev regularity of the p-LaplaceLocal Holder regularity of the p-Laplace equation
Besov regularity of solutions of the p-Laplace equationFrom Bs
p,p(Ω) and C `,αγ,loc(Ω) to Bσ
τ,τ (Ω)Besov regularity of the p-Laplace
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 11 of 23
Sobolev and local Holder regularity – Sobolev regularity of the p-Laplace
Sobolev regularity of the p-Laplace
Theorem (Ebmeyer 2001, 2002, Savare 1998)
Ω ⊂ Rd bounded polyhedral domain, d ≥ 2, 1 < p <∞, f ∈ Lp′(Ω).If ∆pu = f and u = 0 on ∂Ω, then
V := |∇u|p−2
2 ∇u ∈W1/2−ε2 (Ω) for all ε > 0 (2)
Furthermore
|∇u| ∈ Lq(Ω) for q <pd
d − 1
and
u ∈
W
3/2−εp (Ω), if 1 < p ≤ 2,
W1+1/p−εp (Ω), if p ≥ 2,
p =p
1− 2−p2d
> p.
Open question: Does (2) hold for general Lipschitz domains?C. Ebmeyer. Nonlinear elliptic problems with p-structure under mixed boundary valueconditions in polyhedral domains. Adv. Diff. Equ., 6:873–895, 2001.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 12 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Local Holder regularity of the homogen. p-Laplace
Replacement for the local (weighted) Sobolev regularity (p = 2)
Theorem (Lewis 1983; Ural’ceva; Evans; DiBenedetto;. . .)
Ω ⊂ Rd bounded open set, d ≥ 2, 1 < p <∞. There exists α ∈ (0, 1]s.t. u with ∆pu = 0 fulfils: ∀ x0 ∈ Ω, r > 0 s.t. B(x0, 64r) ⊂ Ω
maxx∈B(x0,r)
|∇u(x)| ≤ C
( ∫−
B(x0,32r)|∇u|pdx
)1/p
≤ C · r−d/p,
maxx ,y∈B(x0,r)
|∇u(x)−∇u(y)| ≤ C · r−α( ∫−
B(x0,32r)|∇u|pdx
)1/p
|x − y |α.
⇒ local (weighted) Holder regularity for homogeneous p-Laplace
J. Lewis. Regularity of the derivatives of solutions to certain degenerate elliptic equations.
Indiana Univ. Math. J., 32(6):849–858, 1983.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 13 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Local Holder regularity of the homogen. p-Laplace
Replacement for the local (weighted) Sobolev regularity (p = 2)
Theorem (Lewis 1983; Ural’ceva; Evans; DiBenedetto;. . .)
Ω ⊂ Rd bounded open set, d ≥ 2, 1 < p <∞. There exists α ∈ (0, 1]s.t. u with ∆pu = 0 fulfils: ∀ x0 ∈ Ω, r > 0 s.t. B(x0, 64r) ⊂ Ω
maxx∈B(x0,r)
|∇u(x)| ≤ C
( ∫−
B(x0,32r)|∇u|pdx
)1/p
≤ C · r−d/p,
maxx ,y∈B(x0,r)
|∇u(x)−∇u(y)| ≤ C · r−α( ∫−
B(x0,32r)|∇u|pdx
)1/p
|x − y |α.
⇒ local (weighted) Holder regularity for homogeneous p-Laplace
J. Lewis. Regularity of the derivatives of solutions to certain degenerate elliptic equations.
Indiana Univ. Math. J., 32(6):849–858, 1983.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 13 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Local Holder regularity of the inhomog. p-Laplace
We can transfer the local Holder regularity from the homogeneouscase to the inhomogeneous p-Laplace equation:
Theorem (Kuusi,Mingione 2013; Diening,Kaplicky,Schwarzacher)
Ω,d,p as before. Let
α∗ = supα : Theorem of Lewis holds including the estimates.
Then for u with ∆pu = f ∈ C 1,β(α):
u is locally α-Holder continuous for α < min(α∗, 1/(p − 1)).
Analog estimates hold for local Holder-seminorm of u.
Problem: α∗ ∈ (0, 1] is unknown for d ≥ 3. (later: case d = 2)
T. Kuusi and G. Mingione. Guide to Nonlinear Potential Estimates. Bull. Math. Sci,4(1):1–82, 2014.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 14 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Local Holder regularity of the inhomog. p-Laplace
We can transfer the local Holder regularity from the homogeneouscase to the inhomogeneous p-Laplace equation:
Theorem (Kuusi,Mingione 2013; Diening,Kaplicky,Schwarzacher)
Ω,d,p as before. Let
α∗ = supα : Theorem of Lewis holds including the estimates.
Then for u with ∆pu = f ∈ C 1,β(α):
u is locally α-Holder continuous for α < min(α∗, 1/(p − 1)).
Analog estimates hold for local Holder-seminorm of u.
Problem: α∗ ∈ (0, 1] is unknown for d ≥ 3. (later: case d = 2)
T. Kuusi and G. Mingione. Guide to Nonlinear Potential Estimates. Bull. Math. Sci,4(1):1–82, 2014.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 14 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Locally weighted Holder spaces C `,αγ,loc(Ω)
∼ 1
∼ 1‖ ‖C 1,α ∼ 1
∼ 2−1
∼ 2−1∼ 2γ
∼ 2−k
2−k
2kγ
C `,αγ,loc(Ω). . . Holder space,
locally weighted, with
`. . . number of derivatives
α. . . Holder exponent ofderivatives of order `
γ. . . growth of Holder exp.with distance to ∂Ω
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 15 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Locally weighted Holder spaces C `,αγ,loc(Ω) (ii)
Definition (Locally weighted Holder spaces)
K compact subset of Ω, δK distance to ∂Ω, K family of compactsubsets of Ω, g ∈ C `(Ω), set
|g |C `,α(K) :=∑|ν|=`
supx ,y∈K ,
x 6=y
|∂νg(x)− ∂νg(y)||x − y |α
,
|g |C 1,αγ,loc(K)
:= supK∈K
δγK |g |C `,α(K) <∞,
C `,αγ,loc(Ω;K) =
g ∈ C `(Ω) : |g |
C `,αγ,loc(K)<∞
.
K shall be the set of all B(x0, r) such that B(x0, 64r) ⊂ Ω.
This definition (` = 1) is perfectly adapted to Lewis’ Theorem.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 16 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
Local Holder regularity of the p-Laplace
Although the optimal local Holder regularity of the solution of thep-Poisson is unknown (d ≥ 3), we can estimate γ by Lewis’ Theorem
maxx ,y∈B(x0,r)
|∇u(x)−∇u(y)| ≤ C · r−α( ∫−
B(x0,32r)|∇u|pdx
)1/p
|x − y |α
≤ C · r−α( ∫−
B(x0,32r)|∇u|qdx
)1/q
|x − y |α, p ≤ q.
≤ C · r−α−d/q · ‖∇u Lq(Ω)‖ · |x − y |α.
Hence, using the result of Ebmeyer
|∇u| ∈ Lq(Ω) for q <pd
d − 1,
we are allowed to choose
γ = α + (d − 1)/p + ε for all ε > 0.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 17 of 23
Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation
The case d = 2: Holder regularity of the p-Poisson (i)
Theorem (Lindgren, Lindqvist 2013; (DDHSW 2014))
Ω ⊂ R2 bounded polygonal domain, 1 < p <∞, f ∈ L∞(Ω). If∆pu = f , u = 0 on ∂Ω, then u is locally α-Holder continuous for all
α <
1, if 1 < p ≤ 2,
1p−1 , if 2 < p <∞.
Furthermore, for the same α’s, it holds
u ∈ C 1,αγ,loc(Ω) for γ = α + 1/p + ε.
The regularity 1p−1 is a natural bound, take v(x) = |x |p/(p−1).
homogen. case: Iwaniec, Manfredi (1989) proved u ∈ C `,αloc (Ω) with
`+ α = 1 +1
6
(1 +
1
p − 1+
√1 +
14
p − 1+
1
(p − 1)2
)>max
(2,
p
p − 1
)Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 18 of 23
Besov regularity of the p-Laplace equation
Table of contents
Introduction and results for the Laplace equation (p = 2)Introduction to the p-LaplaceApproximation in Sobolev and Besov spacesKnown results for the Laplace equation (p = 2)
Sobolev and local Holder regularity of the p-LaplaceSobolev regularity of the p-LaplaceLocal Holder regularity of the p-Laplace equation
Besov regularity of solutions of the p-Laplace equationFrom Bs
p,p(Ω) and C `,αγ,loc(Ω) to Bσ
τ,τ (Ω)Besov regularity of the p-Laplace
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 19 of 23
Besov regularity of the p-Laplace equation – From Bsp,p (Ω) and C
`,αγ,loc
(Ω) to Bστ,τ (Ω)
From B sp,p(Ω) and C `,α
γ,loc(Ω) to Bστ,τ(Ω)
Theorem (Dahlke, Diening, Hartmann, S., Weimar(DDHSW) ’14)
Ω ⊂ Rd bound. Lipschitz dom., d ≥ 2, s > 0, 1 < p <∞, α ∈ (0, 1],
σ∗ =
`+ α, if 0 < γ < `+α
d + 1p ,
dd−1
(`+ α + 1
p − γ), if `+α
d + 1p ≤ γ < `+ α + 1
p ,
then for all
0 < σ < min
σ∗,
d
d − 1s
and
1
τ=σ
d+
1
p
we have the continuous embedding
Bsp,p(Ω) ∩ C `,α
γ,loc(Ω) → Bστ,τ (Ω).
If γ not too bad and local Holder regularity `+ α is higher thanSobolev regularity s, Besov regularity σ is higher than Sobolev reg. !
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 20 of 23
Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace
The case d = 2: Besov regularity of the p-Poisson
1. By Ebmeyer’s result
u ∈
B
3/2−εp,p (Ω), if 1 < p ≤ 2,
B1+1/p−εp,p (Ω), if p ≥ 2,
2. Lindgren, Lindqvist:
u ∈ C 1,αγ,loc(Ω), γ = α + 1/p + ε, α <
1, if 1 < p ≤ 2,
1p−1 , if 2 < p <∞.
3. γ not too bad? α + 1p + ε = γ
?< `+α
d + 1p = 1+α
2 + 1p ? Yes, α < 1
4. General embedding theorem, 1τ = σ
d + 1p ,
u ∈ Bστ,τ (Ω) for all σ <
2, if 1 < p ≤ 2,
1 + 1p−1 , if 2 < p <∞.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 21 of 23
Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace
Summary: Besov regularity of the p-Poisson
For d = 2 results on Besov regularity beat Sobolev regularity:
it holds
2 > 3/2, if 1 < p ≤ 2,
1 + 1p−1 > 1 + 1
p if 2 < p <∞. For d ≥ 3 the optimal α is unknown, known: α→ 0 for p →∞ For d ≥ 3 to beat Sobolev regularity we need
α >
12 , if 1 < p < 2,1p , if p > 2,
and γ not too large depending on d . This implies
p ∈ (pd ,∞) with pd →∞ for d →∞.
E. Lindgren and P. Lindqvist. Regularity of the p-poisson equation in the plane.arXiv:1311.6795v2, 2013.
T. Iwaniec and J. Manfredi. Regularity of p-harmonic functions on the plane. Rev. Mat.Iberoamericana, 5(1-2):119, 1989.
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 22 of 23
Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace
Open problems
d = 2, can one do better, in dependency of the angles of theboundary?
Is the chosen γ optimal? non-polyhedral domains measure Besov regularity in Lq for the p-Laplace (q 6= p)
bring the F s,rlocp,q (Ω) spaces into play...
work in progress. . .
Thank you for your attention
e-mail: benjamin.scharf@ma.tum.de
web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 23 of 23
Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace
Open problems
d = 2, can one do better, in dependency of the angles of theboundary?
Is the chosen γ optimal? non-polyhedral domains measure Besov regularity in Lq for the p-Laplace (q 6= p)
bring the F s,rlocp,q (Ω) spaces into play...
work in progress. . .
Thank you for your attention
e-mail: benjamin.scharf@ma.tum.de
web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 23 of 23