The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition

The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition Sukjung Hwang CMAC, Yonsei University Collaboration with M. Dindos and M. Mitrea The 1st Meeting of Young Researchers in PDEs August 19, 2016 Hwang Carleson condition 1 / 31

Table of contents 1 Parabolic Equations with L p Dirichlet data Carleson measure and Lipschitz domain Non-tangential maximal functions Pullback Transformation Square and Area functions 2 L 2 Solvability for Parabolic equations 3 Solvaility for Elliptic system Hwang Carleson condition 2 / 31

Elliptic equation Definition of Solvability For 1 < p and Ω (an admissible Lipschitz domain), consider the parabolic Dirichlet boundary value problems 0 = i,j ( x i aij (x) xj u(x) ) in Ω, u = f L p ( Ω) on Ω, (1) N(u) L p ( Ω, dσ). where A satisfies the uniform ellipticity conditions. We say (1) is solvable if there exists a constant C such that N(u) L p ( Ω,dσ) C f L p ( Ω,dσ) where C depends on the operator, p, and the domain. Hwang Carleson condition 3 / 31

Negative Results Theorem Consider 0 = i,j xi ( aij (x) xj u(x) ). There exists a bounded measurable coefficient matrix A = [a ij ] on a unit disk D satisfying the uniform ellipticity conditions such that the L p Dirichlet problem is not solvable for any p (1, ). It is difficult because both the coefficients and the domain are given rough. We impose minimal and natural smoothness conditions (so called Carleson conditions) on the coefficients. Hwang Carleson condition 4 / 31

L p Dirichlet problems Definition of Solvability For 1 < p and Ω (an admissible Lipschitz domain), consider the parabolic Dirichlet boundary value problems u t = div (A u) + B u in Ω, u = f L p ( Ω) on Ω, (2) N(u) L p ( Ω, dσ). where A satisfies the uniform ellipticity conditions. We say (2) is solvable if there exists a constant C such that N(u) L p ( Ω,dσ) C f L p ( Ω,dσ) where C depends on the operator, p, and the domain. Hwang Carleson condition 5 / 31

Parabolic distance and ball Parabolic distance: Parabolic ball: δ(x, t) = ( inf X Y + t s 1/2). (Y,s) Ω B r (X, t) = {(Y, s) R n R : X Y + t s 1/2 < r}. Surface ball and Corresponding Interior: r (X, t) = Ω B r (X, t), T ( r )(X, t) = Ω B r (X, t). Hwang Carleson condition 6 / 31

Carleson Measure Definition Then a measure µ : Ω R + is said to be Carleson if there exists a constant C = C(r 0 ) such that for all r r 0 and all surface balls r µ (T ( r )) Cσ( r ). The best constant C(r 0 ) is called the Carleson norm. If lim r0 0 C(r 0 ) = 0, then we say that the measure µ satisfies the vanishing Carleson condition. Hwang Carleson condition 7 / 31

Assume that δ(x, t) 1 ( osc Bδ(X,t)/2 (X,t)a ij ) 2 + δ(x, t) ( sup Bδ(X,t)/2 (X,t)b i ) 2 is the density of Carleson measure with small Carleson norm. Hwang Carleson condition 8 / 31

Assume that δ(x, t) 1 ( osc Bδ(X,t)/2 (X,t)a ij ) 2 + δ(x, t) ( sup Bδ(X,t)/2 (X,t)b i ) 2 is the density of Carleson measure with small Carleson norm. Small Carleson conditions on the coefficients ( A 2 δ(x, t) + A t 2 δ 3 (X, t) ) dx dt, B 2 δ(x, t) dx dt are Carleson measures with the norm ɛ. Hwang Carleson condition 8 / 31

Lipschitz domain for Elliptic case For X = (x 0, x) R R n 1, consider the region given by Ω = {(x 0, x) : x 0 > ψ(x)} where the function ψ satisfies the followings ψ(x) ψ(y) L x y. Hwang Carleson condition 9 / 31

The time-varying Lipschitz domain For X = (x 0, x) R R n 1, consider the region given by Ω = {(x 0, x, t) : x 0 > ψ(x, t)} where the function ψ satisfies the followings ( ψ(x, t) ψ(y, s) L x y + t s 1/2). Hunt s conjecture: Lip(1, 1/2) is proper domain for parabolic equations Hwang Carleson condition 10 / 31

The admissible time-varying Lipschitz domain For X = (x 0, x) R R n 1, consider the region given by Ω = {(x 0, x, t) : x 0 > ψ(x, t)} where the function ψ satisfies the followings ( ψ(x, t) ψ(y, s) L x y + t s 1/2), D t 1/2 ψ BMO L. Lewis & Murray (1995) : Establish mutual absolute continuity of caloric measure and a certain parabolic analogue of surface measure An admissible parabolic is defined locally satisfying Lip(1, 1/2) and Lewis-Murray conditions using at most N L cylinders. Hwang Carleson condition 11 / 31

Non-tangential maximal functions For (x 0, x, t) Ω, define a parabolic cone Γ a (x 0, x, t) = {(y 0, y, s) Ω : x y + s t 1/2 a(y 0 x 0 ), y 0 > x 0 }. Non-tangential maximal functions: N a (u)(x 0, x, t) = sup u(y 0, y, s). (y 0,y,s) Γ a(x 0,x,t) Hwang Carleson condition 12 / 31

Property of Carleson Measure Non-tangential maximal function and Carleson measure We have integral inequalities that ( δ A 2 + δ 3 A t 2 + δ B 2) u 2 dx dt T ( r ) µ C r N 2 (u) dx dt where µ C is a Carleson norm. Hwang Carleson condition 13 / 31

Dahlberg (1977): Elliptic problem with L 2 Dirichlet boundary data Kenig, Koch, Pipher, Toro (2000): Elliptic operator satisfying Carleson measure condition, Comparison of the square and non-tangential maximal functions Dindos, Petermichl, Pipher (2007): L p Dirichlet Elliptic problems under small Carleson conditions, Quantitative approach Symmetry of the coefficients is NOT assumed. Does NOT rely on layer potentials rather direct approach. Compare the non-tangential maximal function and the square function Hwang Carleson condition 14 / 31

Pullback Transformation Hwang Carleson condition 15 / 31

Pullback Transformation x 0 independent transformation It is trivial to use ρ(x 0, x, t) = (x 0 + ψ(x, t), x, t). But v t provides ψ t (x, t)u x0 (x 0, x, t) where ψ t may not defined because of the lack of regularity. For Elliptic equations, corresponding coefficient A is x 0 independent. (Kenig, Koch, Pipher, & Toro 2000, 2 dimension for Dirichlet problem, Hofmann, Kenig, Mayboroda, & Pipher Higher dimensional Dirichlet problem, Rellich inequality) Hwang Carleson condition 16 / 31

Dalhberg-Kenig-Necas-Stein transformation using mollification We define ρ(x 0, x, t) = (x 0 + P γx0 ψ(x, t), x, t). For a nonnegative function P(x, t) C 0 for (x, t) R n 1 R, set P λ (x, t) λ (n+1) P(λ 1 x, λ 2 t), P λ ψ(x, t) P λ (x y, t s)ψ(y, s) dy ds. R n 1 R Elliptic Cases: basically allows Layer potentials (Mitrea & Taylor 2003, Hofamman & Lewis 2001 Double layer potentials for parabolic equations) We avoid layer potential methods. Hwang Carleson condition 17 / 31

Definitions Square functions: ( 1/2 S a (u)(x 0, x, t) = u 2 δ n (y 0, y, s) dy 0 dy ds), Γ a(x 0,x,t) S a (u) 2 L 2 ( Ω) = u 2 δ(y 0, y, s) dy 0 dy ds Ω Hwang Carleson condition 18 / 31

Definitions Square functions: ( 1/2 S a (u)(x 0, x, t) = u 2 δ n (y 0, y, s) dy 0 dy ds), Γ a(x 0,x,t) S a (u) 2 L 2 ( Ω) = u 2 δ(y 0, y, s) dy 0 dy ds Ω Area functions: ( 1/2 A a (u)(x 0, x, t) = u t 2 δ n+2 (y 0, y, s) dy 0 dy ds), Γ a(x 0,x,t) A a (u) 2 L 2 ( Ω) = u t 2 δ 3 (y 0, y, s) dy 0 dy ds. Ω Hwang Carleson condition 18 / 31

Strategy L : the Maximum Principle L p for 2 < p < : Interpolation if we have L 2 solvability Hwang Carleson condition 19 / 31

Strategy L : the Maximum Principle L p for 2 < p < : Interpolation if we have L 2 solvability L 2 : Quantitative method comparing S(u) L 2 ( Ω) and N(u) L 2 ( Ω) L p for 1 < p < 2: Quantitative method with p adapted square and area functions. Hwang Carleson condition 19 / 31

L 2 Dirichlet problems Definition of Solvability For Ω (an admissible Lipschitz domain), consider the parabolic Dirichlet boundary value problems u t = div (A u) + B u in Ω, u = f L 2 ( Ω) on Ω, (3) N(u) L 2 ( Ω, dσ). where A satisfies the uniform ellipticity conditions. We say (3) is solvable if there exists a constant C such that N(u) L2 ( Ω,dσ) C f L 2 ( Ω,dσ) where C depends on the operator and the domain. Hwang Carleson condition 20 / 31

L 2 Dirichlet problem Uniform Ellipticity and Boundedness for A = [a ij ] λ ξ 2 a ij ξ i ξ j Λ ξ 2. Small Carleson conditions on the coefficients ( A 2 δ(x, t) + A t 2 δ 3 (X, t) ) dx dt, B 2 δ(x, t) dx dt are Carleson measures with the norm ɛ. Boundedness from the Carleson conditions A δ(x, t) + A t δ 2 (x, t) ɛ 1/2, B δ(x, t) ɛ 1/2. Hwang Carleson condition 21 / 31

Rough Sketch It is the results of three estimates. Comparing A(u) and S(u) using Caccioppoli s inequality Hwang Carleson condition 22 / 31

Rough Sketch It is the results of three estimates. Comparing A(u) and S(u) using Caccioppoli s inequality First, Ω S 2 (u) dx dt C Ω f 2 dx dt + ɛ N 2 (u) dx dt Ω where ɛ comes from the smallness of the Carleson measure of the coefficients. Hwang Carleson condition 22 / 31

We replace S(u) L 2 ( Ω) by the first inequality. Then Ω [ N 2 (u) dx dt C C Ω Ω S 2 (u) dx dt + f 2 dx dt + ɛ Ω Ω ] f 2 dx dt N 2 (u) dx dt. Hwang Carleson condition 23 / 31

We replace S(u) L 2 ( Ω) by the first inequality. Then Ω [ N 2 (u) dx dt C C Ω Ω S 2 (u) dx dt + f 2 dx dt + ɛ Because of ɛ, we can hide N(u) L 2 ( Ω). Therefore, it follows Ω N(u) L 2 ( Ω) C f L 2 ( Ω). Ω ] f 2 dx dt N 2 (u) dx dt. Hwang Carleson condition 23 / 31

Theorem For an admissible parabolic domain Ω, let A = [a ij ] satisfy the uniform ellipticity and boundedness with constants λ and Λ. In addition, assume that [ ( ) ] 2 dµ = δ(x, t) 1 sup osc a ij + δ(x, t) sup B 2 dx dt 1 i,j n B δ(x,t)/2 (X,t) B δ(x,t)/2 (X,t) is the density of a Carleson measure on Ω with Carleson norm µ C. Continue.. Hwang Carleson condition 24 / 31

Theorem Then there exists ε > 0 such that if for some r 0 > 0 max{l, µ C,r0 } < ε then the L p boundary value problem u t = div(a u) + B u in Ω, u = f L p on Ω, (4) N(u) L p ( Ω), is solvable for all 1 p <. Moreover, the estimate holds with C p = C p (L, µ C,r0, N, λ, Λ). N(u) Lp ( Ω,dσ) C p f Lp ( Ω,dσ), (5) Hwang Carleson condition 25 / 31

Solvability for Elliptic system Joint work with M. Dindos and M. Mitrea (U of Missouri) Provide quantitative method for Elliptic system with L 2 Dirichlet boundary condition. The lack of regularity such as no Maximum principle. Not always possible to obatin L p solvability for p > 2. With Regularity problem and Reverse Hölder inequality, in fact, we have results on L p Dirichlet problem for 2 p < p where p depending on n and p. Hwang Carleson condition 26 / 31

Definition Let Ω be the Lipschitz domain {(x 0, x ) : x 0 > φ(x )}. Let 1 < p <. Consider the following Dirichlet problem for a vector valued function u : Ω R n ( ) ] 0 = [ i A αβ ij (x) j u β + B αβ i (x) i u β in Ω, α = 1, 2,..., N α u(x) = f (x) on Ω, (6) Ñ(u) L p ( Ω). Here we use the usual Einstein convention and sum over repeating indices (i and β). We say the Dirichlet problem (6) is solvable for a given p (1, ), if there exists C = C(λ, Λ, n, p, Ω) > 0 such that the unique energy solution (u W 1,2 (Ω) obtained via Lax-Milgram lemma) with boundary data f L p ( Ω) B 2,2 1/2 ( Ω) satisfies the estimate Ñu L p ( Ω) C f L p ( Ω). (7) Hwang Carleson condition 27 / 31

Definition For Ω R n as above the nontangential maximal function of u at Q Ω relative to the cone Γ a (Q) and its truncated version at height h are defined by N a (u)(q) = sup u(x) and Na h (u)(q) = sup u(x). x Γ a(q) x Γ h a (Q) Moreover, we shall also consider a weaker version Ñ of the nontangential maximal function that is defined using L 2 averages over balls in the domain Ω. We set Ñ a (u)(q) = sup w(x) and Ña h (u)(q) = sup w(x) x Γ a(q) x Γ h a (Q) where ( 1 w(x) := B δ(x)/2 (x) B δ(x)/2 (x) u 2 (z) dz) 1/2. (8) Hwang Carleson condition 28 / 31

Theorem (assumptions) Let Ω be the Lipschitz domain {(x 0, x ) : x 0 > φ(x )} with Lipschitz constant L = φ L. Assume that the coefficients A of the system (6) are strongly elliptic with parameters λ, Λ. In addition assume that at least one of the following holds. ( (i) Ω = R n + and the matrix A 00 = (ii) For all i, j, α, β we have A αβ ij A αβ 00 ) α,β = A βα ij on Ω. = I N N on Ω. Hwang Carleson condition 29 / 31

Theorem (Results) Then there exists K = K(λ, Λ, n) > 0 such that if ( ( dµ(x) = 2 sup A(x) ) + B δ(x)/2 (x) sup B(x) B δ(x)/2 (x) ) 2 δ(x) dx (9) is a Carleson measure with norm µ C less than K and the Lipschitz constant L < K the the L 2 Dirichlet problem for the system (6) is solvable and the estimate Ñu L 2 ( Ω) C f L 2 ( Ω). holds for all energy solutions u with f L 2 ( Ω) B 2,2 1/2 ( Ω) where C(λ, Λ, n, Ω) > 0. Hwang Carleson condition 30 / 31

Thank you for your attention!! Hwang Carleson condition 31 / 31