Besov regularity of solutions of the p-Laplace equation
Benjamin Scharf
Technische Universitat Munchen,Department of Mathematics,Applied Numerical Analysis
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: [email protected]
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: [email protected]
web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf
Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 23 of 23