Theory of Partial Differential Equations in Sobolev spaces: Perspectives and Developments Tuoc Phan University of Tennessee, Knoxville, TN Pure Mathematics Colloquium: Current Advances in Mathematics Department of Mathematics and Statistics Texas Tech University September, 28, 2020 Support from Simons foundation is gratefully acknowledged T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Outline Introduction: PDE in Sobolev spaces (a) Theory for Laplace equation (Calder´ on-Zygmund) (b) Some extension: known results for equations with uniformly elliptic and bounded coefficients Equations with singular-degenerate coefficients: motivations, problems/questions Some (simplified) results for equations with singular or degenerate coefficients Ideas in the proofs and remarks T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Linear second order elliptic/parabolic equations Non-divergence form equations u t − a ij ( t , x ) D ij u ( t , x ) = f ( t , x ) Divergence form equations u t − D i [ a ij ( t , x ) D j u ] = f ( t , x ) Here, u ( t , x ) be an unknown physical/biological quantity, f ( t , x ) is a given “external force”, and a ij ( t , x ) . Moreover D i is the spatial partial derivative in i th -direction: D i u = u x i , D ij = u x i x j . Question: If f ∈ L p (( 0 , T ) × Ω) with some p ∈ ( 1 , ∞ ) and some spatial domain Ω ⊂ R d , i.e. � ˆ T � 1 / p ˆ | f ( t , x ) | p dxdt � f � L p (( 0 , T ) × Ω)) = < ∞ Ω 0 can we control u , Du , D 2 u , and u t in L p ? T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Laplace equation We consider the Laplace equation x ∈ R d − ∆ u ( x ) = f ( x ) for where ∆ u = u x 1 x 1 + u x 2 x 2 + · · · + u x d x d For p ∈ ( 1 , ∞ ) , is there N = N ( d , p ) > 0 ˆ ˆ R d | D 2 u ( x ) | p dx ≤ N R d | f ( x ) | p dx for every smooth, compactly supported solution u ? Note that D 2 u is the Hessian matrix of u : D 2 u = ( D ij u ) d ∆ u = trace D 2 u , mean while i , j = 1 where D i u = u x i and D ij u = u x i x j . Note : For an arbitrary matrix M , knowing the trace does not mean we know the matrix. T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Laplace equation: energy estimate ( p = 2) Is it true that ˆ ˆ R d | D 2 u ( x ) | 2 dx ≤ N R d | f ( x ) | 2 dx when − ∆ u ( x ) = f ( x )? By squaring the equations, we obtain d ˆ ˆ � R d | f ( x ) | 2 dx . R d u x i x i ( x ) u x k x k ( x ) dx = i , k = 1 Note that by using the integration by parts, we obtain ˆ ˆ ˆ R d | u x i x k ( x ) | 2 dx . R d u x i x i u x k x k dx = R d u x i x k u x i x k dx = Therefore, ˆ ˆ R d | D 2 u ( x ) | 2 dx = R d | f ( x ) | 2 dx . Question: What about p � 2? T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Calder´ on-Zygmund theory (for Laplace equation) Theorem (Calder´ on-Zygmund (1950-1960)) If u ∈ C ∞ 0 ( R d ) is a solution of x ∈ R d − ∆ u ( x ) = f ( x ) , with f ∈ L p ( R d ) for p ∈ ( 1 , ∞ ) , then ˆ ˆ R d | D 2 u ( x ) | p dx ≤ N ( d , p ) R d | f ( x ) | p dx . Proof. Write ˆ u x i x j ( x ) = R d K ij ( x − y ) f ( y ) dy with some singular kernel K ij . Use their developed “theory of singular integral operators”. � T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Modern approaches Krylov ( ∼ 2003): Based on Fefferman-Stein theorem for sharp function � f # � L p ( R d ) ∼ � f � L p ( R d ) where 1 ˆ f # ( x ) = sup | f ( y ) − f B r ( x ) | dy | B r ( x ) | r > 0 B r ( x ) with f B r ( x ) the average of f in the ball B r ( x ) . Use the PDE to control ( D 2 u ) # Caffarelli-Peral (CPAM - 2003): Based on level set estimates ˆ ∞ ˆ � p d λ R d | D 2 u ( x ) | p dx = N ( n , p ) λ p − 1 � � { x : | D 2 u ( x ) | > λ } � � � 0 � � � { x : | D 2 u ( x ) | > λ } Use the PDE to control � � � Both approaches work for linear, nonlinear and fully nonlinear equations. T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Main steps in Krylov’s approach (oscillation estimates) Let us consider the harmonic function − ∆ u = 0 B 2 ( x 0 ) . in We know (1 st PDE course) � D k u � L ∞ ( B 1 ( x 0 )) ≤ N ( d , k ) | u ( x ) | dx . B 2 ( x 0 ) Then, by mean value theorem (Cal I) | D 2 u ( y ) − D 2 u B 1 ( x 0 ) | dy ≤ � D 3 u � L ∞ ( B 1 ( x 0 )) B 1 ( x 0 ) Therefore, we can control the oscillation of D 2 u in B 1 ( x 0 ) by | D 2 u ( y ) − D 2 u B 1 ( x 0 ) | dy ≤ N ( d ) | u ( x ) | dx . B 1 ( x 0 ) B 2 ( x 0 ) Then (with a little bit more work) we use Fefferman-Stein theorem to derive the L p -estimate of D 2 u . T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Equations with variable coefficients CZ theory has been extended equations in non-divergence and divergence form (elliptic, parabolic, linear, nonlinear) ( ND ) − a ij ( x ) D ij u ( x ) + c ( x ) u ( x ) = f ( x ) ( D ) − D i [ a ij ( x ) D j u ( x )] + c ( x ) u ( x ) = f ( x ) The coefficients matrix a ij is bounded, and uniformly elliptic: there is ν ∈ ( 0 , 1 ) such that ν | ξ | 2 ≤ a ij ( x ) ξ i ξ j | a ij ( x ) | ≤ ν − 1 and for all x and for all ξ = ( ξ 1 , ξ 2 , . . . , ξ d ) ∈ R d . ( a ij ) is sufficiently smooth: a ij ∈ VMO is sufficient (Sarason’s class of functions). Refs: Di Fazio-Ragusa (1991); Maugeri-Palagachev-Softova (2000-book); Krylov (2003-book); Caffarelli-Peral (2003); Acerbi-Mingione (2007); Byun-Wang (2012,...), Hoang-Nguyen-P . (2015),... T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
A simple example (for the perturbation technique) Consider R d . − a ij ( x ) D ij u = f in Assume (for simplification) that | a ij ( x ) − δ ij | ≤ ǫ for all x . We write (freezing the coefficients) − ∆ u = g , g := [ a ij ( x ) − δ ij ] D ij u + f Then, � � � D 2 u � L p ( R d ) ≤ N ǫ � D 2 u � L p ( R d ) + � f � L p ( R d ) . If ǫ is sufficiently small, we obtain � D 2 u � L p ( R d ) ≤ N � f � L p ( R d ) . T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Equations with singular/degenerate coefficients + = R d − 1 × ( 0 , ∞ ) the upper half space. We write Denote R d x = ( x ′ , x d ) ∈ R d + where x ′ ∈ R d − 1 and x d ∈ R + = ( 0 , ∞ ) . We study the following class of equations: x α d ( u t + u ) − D i [ x α d a ij ( t , x ) D j u ] = x α ( D ) d f ( t , x ) or u t + u − a ij ( t , x ) D ij u ( t , x ) + α ( ND ) a dj ( t , x ) D j u = f ( t , x ) x d Boundary condition on ∂ R d + = { x d = 0 } : x d → 0 + x α Conormal ( zero flux ) : lim d a dj ( t , x ) D j u ( t , x ) = 0 or u ( t , x ′ , 0 ) = 0 Dirichlet : Here, α ∈ R is given constant; ( a ij ) is bounded, and uniformly elliptic. T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Equations with singular/degenerate coefficients Recall x = ( x ′ , x d ) ∈ R d + , t ∈ R and we consider x α d ( u t + u ) − D i [ x α d a ij ( t , x ) D j u ] = x α ( D ) d f ( t , x ) When α > 0, the coefficients are degenerate. Meanwhile, when α < 0 the coefficients are singular (on { x d = 0 } = ∂ R d + ). Motivation: Geometric PDE, Calculus of Variations, Probability theory, Mathematical finance, Math Biology, Non-local PDE. Question: L p -theory for the PDE (what is the right functional space setting for the PDE?). Issue: The coefficients x α d a ij ( t , x ) may not bounded, not uniformly elliptic, and not sufficiently smooth (even not locally integrable as α ≤ − 1). T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
Energy estimate (the first try) We consider a solution u ∈ C ∞ 0 of the PDE x ∈ R d x α d ( u t + u ) − D i [ x α d a ij ( t , x ) D j u ] = x α d f ( t , x ) + , t ∈ R with either the Dirichlet or the conormal boundary condition on { x d = 0 } . Energy estimate (integration by parts, and Cauchy-Schwartz inequality): ˆ ˆ | Du ( t , x ) | 2 x α | u ( t , x ) | 2 x α d dxdt + d dxdt R d + 1 R d + 1 + + ˆ | f ( t , x ) | 2 x α ≤ N d dxdt . R d + 1 + Common thought: Upgrade this estimate: replacing 2 by p for p ∈ ( 1 , ∞ ) ? T. Phan (UTK) Pure Math. Colloquium Texas Tech, Sept. 28, 2010 PDE in Sobolev spaces: Perspectives and Developments
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