Elliptic boundary value problems with complex coefficients and - PowerPoint PPT Presentation

Elliptic boundary value problems with complex coefficients and fractional regularity data (the first order approach) Alex Amenta (joint work with Pascal Auscher) Delft University of Technology, Netherlands August 18, 2017 BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 1 / 17

The divergence form elliptic equation u : R 1+ n div A ∇ u = 0 , → C . + BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 2 / 17

The divergence form elliptic equation u : R 1+ n div A ∇ u = 0 , → C . + The coefficients A ( t, x ) = A ( x ) ∈ L ∞ ( R n : L ( C 1+ n )) are t -independent, BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 2 / 17

The divergence form elliptic equation u : R 1+ n div A ∇ u = 0 , → C . + The coefficients A ( t, x ) = A ( x ) ∈ L ∞ ( R n : L ( C 1+ n )) are t -independent, uniformly elliptic: there exists κ > 0 such that Re( A ( x ) v, v ) ≥ κ | v | 2 ∀ v ∈ C 1+ n , x ∈ R n , BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 2 / 17

The divergence form elliptic equation u : R 1+ n div A ∇ u = 0 , → C . + The coefficients A ( t, x ) = A ( x ) ∈ L ∞ ( R n : L ( C 1+ n )) are t -independent, uniformly elliptic: there exists κ > 0 such that Re( A ( x ) v, v ) ≥ κ | v | 2 ∀ v ∈ C 1+ n , x ∈ R n , not assumed to be real, symmetric, or smooth in any way. BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 2 / 17

The divergence form elliptic equation u : R 1+ n div A ∇ u = 0 , → C . + The coefficients A ( t, x ) = A ( x ) ∈ L ∞ ( R n : L ( C 1+ n )) are t -independent, uniformly elliptic: there exists κ > 0 such that Re( A ( x ) v, v ) ≥ κ | v | 2 ∀ v ∈ C 1+ n , x ∈ R n , not assumed to be real, symmetric, or smooth in any way. Maximum principle, existence of fundamental solutions, local H¨ older regularity of solutions all fail . BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 2 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  T p θ : weighted tent space (definition on next slide) BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  T p θ : weighted tent space (definition on next slide) H p ˙ θ : homogeneous Hardy–Sobolev space of order θ BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  T p θ : weighted tent space (definition on next slide) H p ˙ θ : homogeneous Hardy–Sobolev space of order θ tangential gradient : ∇ � u = ( ∂ 1 u, . . . , ∂ n u ) BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  T p θ : weighted tent space (definition on next slide) H p ˙ θ : homogeneous Hardy–Sobolev space of order θ tangential gradient : ∇ � u = ( ∂ 1 u, . . . , ∂ n u ) A -conormal derivative : ∂ ν A u = e 0 · A ∇ u ( e 0 : unit vector in the t -direction). BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  T p θ : weighted tent space (definition on next slide) H p ˙ θ : homogeneous Hardy–Sobolev space of order θ tangential gradient : ∇ � u = ( ∂ 1 u, . . . , ∂ n u ) A -conormal derivative : ∂ ν A u = e 0 · A ∇ u ( e 0 : unit vector in the t -direction). Say ( N ) p θ,A is well-posed if for every boundary data ∂ ν A f there exists a unique solution u satisfying the given conditions. BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Boundary value problems For θ ∈ [ − 1 , 0) and p > 1 , formulate the Neumann problem in R 1+ n  div A ∇ u = 0 , +   ||∇ u || T p θ � || ∂ ν A f || ˙ θ ,  H p ( N ) p θ,A : in ( S ′ / P )( R n : C n ) , lim t →∞ ∇ � u ( t, · ) = 0  θ ( R n : C ) .  lim t → 0 ∂ ν A u ( t, · ) = ∂ ν A f ∈ ˙ H p  T p θ : weighted tent space (definition on next slide) H p ˙ θ : homogeneous Hardy–Sobolev space of order θ tangential gradient : ∇ � u = ( ∂ 1 u, . . . , ∂ n u ) A -conormal derivative : ∂ ν A u = e 0 · A ∇ u ( e 0 : unit vector in the t -direction). Say ( N ) p θ,A is well-posed if for every boundary data ∂ ν A f there exists a unique solution u satisfying the given conditions. Goal: find a useful characterisation of well-posedness of ( N ) p θ,A . BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 3 / 17

Weighted tent spaces � � ∞ � p/ 2 � 1 /p � � | t − θ F ( t, y ) | 2 dy dt � || F || T p θ := dx . t n +1 R n 0 B ( x,t ) BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 4 / 17

Weighted tent spaces � � ∞ � p/ 2 � 1 /p � � | t − θ F ( t, y ) | 2 dy dt � || F || T p θ := dx . t n +1 R n 0 B ( x,t ) a solution u to ( N ) p θ,A with boundary data ∂ ν A f must satisfy ||∇ u || T p θ � || ∂ ν A f || ˙ θ . H p BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 4 / 17

Weighted tent spaces � � ∞ � p/ 2 � 1 /p � � | t − θ F ( t, y ) | 2 dy dt � || F || T p θ := dx . t n +1 R n 0 B ( x,t ) a solution u to ( N ) p θ,A with boundary data ∂ ν A f must satisfy ||∇ u || T p θ � || ∂ ν A f || ˙ θ . H p History of tent spaces: BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 4 / 17

Weighted tent spaces � � ∞ � p/ 2 � 1 /p � � | t − θ F ( t, y ) | 2 dy dt � || F || T p θ := dx . t n +1 R n 0 B ( x,t ) a solution u to ( N ) p θ,A with boundary data ∂ ν A f must satisfy ||∇ u || T p θ � || ∂ ν A f || ˙ θ . H p History of tent spaces: Unweighted tent spaces ( θ = 0 ): Coifman–Meyer–Stein 1985. BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 4 / 17

Weighted tent spaces � � ∞ � p/ 2 � 1 /p � � | t − θ F ( t, y ) | 2 dy dt � || F || T p θ := dx . t n +1 R n 0 B ( x,t ) a solution u to ( N ) p θ,A with boundary data ∂ ν A f must satisfy ||∇ u || T p θ � || ∂ ν A f || ˙ θ . H p History of tent spaces: Unweighted tent spaces ( θ = 0 ): Coifman–Meyer–Stein 1985. First definition with θ � = 0 : Hofmann–Mayboroda–McIntosh 2011. BVP / L ∞ coefficients / fractional regularity data Alex Amenta (TU Delft) August 18, 2017 4 / 17