Representation for weak solutions of elliptic boundary value problems . Auscher 1 P 1 Université Paris-Sud, France Workshop on harmonic analysis, PDE and geometric measure theory ICMAT, 12-16 january 2015 P . Auscher Representation
based on two joint works with my student Sebastian Stahlhut and with Mihalis Mourgoglou, available on arXiv. development of Dirac operators for BVP from earlier works with Andreas Axelsson, Alan McIntosh, Steve Hofmann. Nothing could be done without the methodology of the solution of the Kato conjecture. P . Auscher Representation
Systems Ω = R n + 1 . Same analysis works in unit ball and every domain + obtained by bilipschitz change of variables. Points in Ω : ( t , x ) , t > 0 , x ∈ R n . Measurable, bounded, with M m × m ( C ) -valued coefficients A i , j , i , j = 0 , . . . n , m ≥ 1. + Ellipticity (later) Weak solution: u ∈ W 1 , 2 loc (Ω; C m ) and Lu = 0 holds in D ′ (Ω; C m ) : with summation convention � i , j ∂ j u β ∂ i ϕ α dxdt = 0 , A α,β ∀ ϕ ∈ C ∞ 0 (Ω; C m ) . Re Ω i , j ∂ j u β ∂ i ϕ α = A ∇ u · ∇ ϕ and Lu = div A ∇ u in Short notation: A α,β Ω . i = 0 corresponds to the vertical direction, i = 1 , . . . , n to the horizontal directions. P . Auscher Representation
Strongly elliptic real equations • local regularity theory (Nash-Moser) • Maximum principle: the classical Dirichlet problem with data f ∈ C c ( R n ) can be uniquely solved: u ∈ C (Ω) is bounded with � u � ∞ ≤ � f � ∞ and can be represented by applying the Riesz representation theorem: � R n f d ω t , x u ( t , x ) = L Probability measure ω t , x is the L -harmonic measure for L at L pole ( t , x ) . • Possible ansatz by using layer potential methods from the fundamental solution. • Many results starting in the late ’70s for real symmetric equations: Dahlberg, Jerison, Kenig, Verchota, R. Fefferman, Pipher.... and recently for real non-symmetric equations: Hofmann, Kenig, Mayboroda, Pipher. P . Auscher Representation
Strongly elliptic real equations • local regularity theory (Nash-Moser) • Maximum principle: the classical Dirichlet problem with data f ∈ C c ( R n ) can be uniquely solved: u ∈ C (Ω) is bounded with � u � ∞ ≤ � f � ∞ and can be represented by applying the Riesz representation theorem: � R n f d ω t , x u ( t , x ) = L Probability measure ω t , x is the L -harmonic measure for L at L pole ( t , x ) . • Possible ansatz by using layer potential methods from the fundamental solution. • Many results starting in the late ’70s for real symmetric equations: Dahlberg, Jerison, Kenig, Verchota, R. Fefferman, Pipher.... and recently for real non-symmetric equations: Hofmann, Kenig, Mayboroda, Pipher. P . Auscher Representation
Strongly elliptic real equations • local regularity theory (Nash-Moser) • Maximum principle: the classical Dirichlet problem with data f ∈ C c ( R n ) can be uniquely solved: u ∈ C (Ω) is bounded with � u � ∞ ≤ � f � ∞ and can be represented by applying the Riesz representation theorem: � R n f d ω t , x u ( t , x ) = L Probability measure ω t , x is the L -harmonic measure for L at L pole ( t , x ) . • Possible ansatz by using layer potential methods from the fundamental solution. • Many results starting in the late ’70s for real symmetric equations: Dahlberg, Jerison, Kenig, Verchota, R. Fefferman, Pipher.... and recently for real non-symmetric equations: Hofmann, Kenig, Mayboroda, Pipher. P . Auscher Representation
Complex equations or systems • no local regularity • no maximum principle • no fundamental solution P . Auscher Representation
BVP problems in L p , 1 < p < ∞ Typical problems in harmonic analysis (for example for the Laplace equation). • (Dir, A, p): Solve Lu = 0 with � � N ( u ) � p < ∞ and u 0 = f given in L p ( R n ; C m ) . • (Reg, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∇ tan u 0 = ∇ tan f , f given in ˙ W 1 , p ( R n ; C m ) . • (Neu, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∂ ν A u | t = 0 = g given in L p ( R n ; C m ) . � N ( h ) is non-tangential maximal interior control of h defined in Ω : it comes up quite naturally. Not always solvable nor well-posed. No comprehensive theory at this time. Find trace and representation not on u but on its full gradient. P . Auscher Representation
BVP problems in L p , 1 < p < ∞ Typical problems in harmonic analysis (for example for the Laplace equation). • (Dir, A, p): Solve Lu = 0 with � � N ( u ) � p < ∞ and u 0 = f given in L p ( R n ; C m ) . • (Reg, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∇ tan u 0 = ∇ tan f , f given in ˙ W 1 , p ( R n ; C m ) . • (Neu, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∂ ν A u | t = 0 = g given in L p ( R n ; C m ) . � N ( h ) is non-tangential maximal interior control of h defined in Ω : it comes up quite naturally. Not always solvable nor well-posed. No comprehensive theory at this time. Find trace and representation not on u but on its full gradient. P . Auscher Representation
BVP problems in L p , 1 < p < ∞ Typical problems in harmonic analysis (for example for the Laplace equation). • (Dir, A, p): Solve Lu = 0 with � � N ( u ) � p < ∞ and u 0 = f given in L p ( R n ; C m ) . • (Reg, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∇ tan u 0 = ∇ tan f , f given in ˙ W 1 , p ( R n ; C m ) . • (Neu, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∂ ν A u | t = 0 = g given in L p ( R n ; C m ) . � N ( h ) is non-tangential maximal interior control of h defined in Ω : it comes up quite naturally. Not always solvable nor well-posed. No comprehensive theory at this time. Find trace and representation not on u but on its full gradient. P . Auscher Representation
BVP problems in L p , 1 < p < ∞ Typical problems in harmonic analysis (for example for the Laplace equation). • (Dir, A, p): Solve Lu = 0 with � � N ( u ) � p < ∞ and u 0 = f given in L p ( R n ; C m ) . • (Reg, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∇ tan u 0 = ∇ tan f , f given in ˙ W 1 , p ( R n ; C m ) . • (Neu, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∂ ν A u | t = 0 = g given in L p ( R n ; C m ) . � N ( h ) is non-tangential maximal interior control of h defined in Ω : it comes up quite naturally. Not always solvable nor well-posed. No comprehensive theory at this time. Find trace and representation not on u but on its full gradient. P . Auscher Representation
BVP problems in L p , 1 < p < ∞ Typical problems in harmonic analysis (for example for the Laplace equation). • (Dir, A, p): Solve Lu = 0 with � � N ( u ) � p < ∞ and u 0 = f given in L p ( R n ; C m ) . • (Reg, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∇ tan u 0 = ∇ tan f , f given in ˙ W 1 , p ( R n ; C m ) . • (Neu, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∂ ν A u | t = 0 = g given in L p ( R n ; C m ) . � N ( h ) is non-tangential maximal interior control of h defined in Ω : it comes up quite naturally. Not always solvable nor well-posed. No comprehensive theory at this time. Find trace and representation not on u but on its full gradient. P . Auscher Representation
BVP problems in L p , 1 < p < ∞ Typical problems in harmonic analysis (for example for the Laplace equation). • (Dir, A, p): Solve Lu = 0 with � � N ( u ) � p < ∞ and u 0 = f given in L p ( R n ; C m ) . • (Reg, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∇ tan u 0 = ∇ tan f , f given in ˙ W 1 , p ( R n ; C m ) . • (Neu, A, p): Solve Lu = 0 with � � N ( ∇ u ) � p < ∞ and ∂ ν A u | t = 0 = g given in L p ( R n ; C m ) . � N ( h ) is non-tangential maximal interior control of h defined in Ω : it comes up quite naturally. Not always solvable nor well-posed. No comprehensive theory at this time. Find trace and representation not on u but on its full gradient. P . Auscher Representation
non-tangential maximal function Whitney ball: W ( t , x ) := [( 1 − c 0 ) t , ( 1 + c 0 ) t ] × B ( x ; c 1 t ) , for fixed c 0 ∈ ( 0 , 1 ) , c 1 > 0. � t − ( n + 1 ) / 2 � h � L 2 ( W ( t , x )) N ( h )( x ) := sup t > 0 It is the L 2 -variant of the usual pointwise maximal function h ∗ ( x ) = sup | h ( t , y ) | . | x − y | < t P . Auscher Representation
Classical Dirichlet problem Theory for L p , 1 < p < ∞ well-known from Fatou type results. Fefferman-Stein extended this to p ≤ 1 using the real Hardy space H p (which agrees with L p when p > 1). Theorem Let 0 < p < ∞ . Let u be harmonic in Ω . The following are equivalent � u ∗ � p < ∞ . 1 � S ( t ∇ u ) � p < ∞ and u vanishes as t → ∞ . 2 There exists a unique f ∈ H p such that u ( t , x ) = P t ∗ f ( x ) , 3 where P t is the Poisson kernel. Moreover, � f � H p ∼ � u ∗ � p ∼ � S ( t ∇ u ) � p . � �� � � � 1 � 2 dtdy � F ( t , y ) Lusin area functional: S ( F )( x ) = 2 | x − y | < t t n + 1 P . Auscher Representation
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