The Neumann problem for symmetric higher order elliptic differential equations Ariel Barton Joint work with Steve Hofmann and Svitlana Mayboroda May 30, 2018 Workshop on Real Harmonic Analysis and its Applications to Partial Differential Equations and Geometric Measure Theory: on the occasion of the 60th birthday of Steve Hofmann I CMAT, Madrid (Spain) Ariel Barton The higher-order Neumann problem May 30, 2018 1 / 39
Second order differential equations: ∆ = @ xx + @ yy + : : : The force required to bend a string under tension is proportional to the second derivative of its displacement, @ xx h . @ tt h = c@ xx h @ xx h = c @ xx h = 0 The force required to bend a membrane under tension is proportional to ∆ h = @ xx h + @ yy h . @ tt h = c ∆ h ∆ h = c ∆ h = 0 Ariel Barton The higher-order Neumann problem May 30, 2018 2 / 39
Harmonic boundary value problems There is an extensive theory for the harmonic Dirichlet problem ( ∆ u = 0 in Ω ; u = f on @ Ω and the Neumann problem ( ∆ u = 0 in Ω ; � · ∇ u = g on @ Ω : Ariel Barton The higher-order Neumann problem May 30, 2018 3 / 39
Second order boundary value problems Suppose the matrix A is uniformly positive definite and bounded: v | 2 ; for all X ∈ R d ; ~ v ∈ C d : Re ~ v · A ( X ) ~ v ≥ – | ~ | A ( X ) | ≤ Λ There is an extensive theory for the second order elliptic Dirichlet problem ( ∇ · A ∇ u = 0 in Ω ; u = f on @ Ω and the Neumann problem ( ∇ · A ∇ u = 0 in Ω ; � · A ∇ u = g on @ Ω : Ariel Barton The higher-order Neumann problem May 30, 2018 4 / 39
Higher order differential equations The force required to bend a thin elastic rod is proportional to the fourth derivative of its displacement, @ xxxx h . The force required to bend a thin elastic plate is proportional to ∆ 2 h = @ xx ( @ xx h ) + @ xy (2 @ xy h ) + @ yy ( @ yy h ). Ariel Barton The higher-order Neumann problem May 30, 2018 5 / 39
Higher order differential equations The force required to bend a thin elastic rod is proportional to the fourth derivative of its displacement, @ xxxx h . The force required to bend a thin elastic plate is proportional to ∆ 2 h = @ xx ( @ xx h ) + @ xy (2 @ xy h ) + @ yy ( @ yy h ). (Euler-Bernoulli beam equation) The force required to bend an inhomogeneous thin elastic rod is proportional to the fourth derivative of its displacement @ xx ( E ( x ) I ( x ) @ xx h ). Ariel Barton The higher-order Neumann problem May 30, 2018 5 / 39
Higher order boundary value problems We are interested in higher-order differential equations such as the biharmonic equation (in R d ) d d X X ∆ 2 u = ∇ 2 · ∇ 2 u = @ jk ( @ jk u ) = 0 j =1 k =1 or more generally X ∇ m · A ∇ m u = @ ¸ ( A ¸˛ @ ˛ u ) = 0 : | ¸ | = | ˛ | = m Ariel Barton The higher-order Neumann problem May 30, 2018 6 / 39
Higher order boundary value problems We are interested in higher-order differential equations such as the biharmonic equation (in R d ) d d X X ∆ 2 u = ∇ 2 · ∇ 2 u = @ jk ( @ jk u ) = 0 j =1 k =1 or more generally X ∇ m · A ∇ m u = @ ¸ ( A ¸˛ @ ˛ u ) = 0 : | ¸ | = | ˛ | = m We are interested in the Dirichlet problem ( ∆ 2 u = 0 in Ω ; u = f ; � · ∇ u = g on @ Ω : Ariel Barton The higher-order Neumann problem May 30, 2018 6 / 39
Higher order boundary value problems We are interested in higher-order differential equations such as the biharmonic equation (in R d ) d d X X ∆ 2 u = ∇ 2 · ∇ 2 u = @ jk ( @ jk u ) = 0 j =1 k =1 or more generally X ∇ m · A ∇ m u = @ ¸ ( A ¸˛ @ ˛ u ) = 0 : | ¸ | = | ˛ | = m We are interested in the Dirichlet problem ( ∆ 2 u = 0 in Ω ; ∇ u = ~ f on @ Ω : Ariel Barton The higher-order Neumann problem May 30, 2018 6 / 39
Higher order boundary value problems We are interested in higher-order differential equations such as the biharmonic equation (in R d ) d d X X ∆ 2 u = ∇ 2 · ∇ 2 u = @ jk ( @ jk u ) = 0 j =1 k =1 or more generally X ∇ m · A ∇ m u = @ ¸ ( A ¸˛ @ ˛ u ) = 0 : | ¸ | = | ˛ | = m We are interested in the Dirichlet problem ( ∇ m · A ∇ m u = 0 in Ω ; ∇ m − 1 u = ˙ f on @ Ω : Ariel Barton The higher-order Neumann problem May 30, 2018 6 / 39
Higher order Neumann boundary values I n the second-order case ∇ · A ∇ u = 0, the Neumann boundary values of u are � · A ∇ u . Ariel Barton The higher-order Neumann problem May 30, 2018 7 / 39
Higher order Neumann boundary values I n the second-order case ∇ · A ∇ u = 0, the Neumann boundary values of u are � · A ∇ u . Notice that if ∇ m · A ∇ m u = 0 in Ω, then ˆ ∇ m ’ · A ∇ m u Ω ˛ depends only on ∇ m − 1 ’ ˛ @ Ω . Ariel Barton The higher-order Neumann problem May 30, 2018 7 / 39
Higher order Neumann boundary values I n the second-order case ∇ · A ∇ u = 0, the Neumann boundary values of u are � · A ∇ u . Notice that if ∇ m · A ∇ m u = 0 in Ω, then ˆ ∇ m ’ · A ∇ m u Ω ˛ depends only on ∇ m − 1 ’ ˛ @ Ω . So ˆ ˆ ∇ m − 1 ’ · ˙ ∇ m ’ · A ∇ m u = M A Ω u dff Ω @ Ω for some ˙ M A Ω u . Ariel Barton The higher-order Neumann problem May 30, 2018 7 / 39
Higher order Neumann boundary values I n the second-order case ∇ · A ∇ u = 0, the Neumann boundary values of u are � · A ∇ u . Notice that if ∇ m · A ∇ m u = 0 in Ω, then ˆ ∇ m ’ · A ∇ m u Ω ˛ depends only on ∇ m − 1 ’ ˛ @ Ω . So ˆ ˆ ∇ m − 1 ’ · ˙ ∇ m ’ · A ∇ m u = M A Ω u dff Ω @ Ω for some ˙ M A Ω u . I f m = 1 then M A Ω u = � · A ∇ u . Ariel Barton The higher-order Neumann problem May 30, 2018 7 / 39
Higher order Neumann boundary values I n the second-order case ∇ · A ∇ u = 0, the Neumann boundary values of u are � · A ∇ u . Notice that if ∇ m · A ∇ m u = 0 in Ω, then ˆ ∇ m ’ · A ∇ m u Ω ˛ depends only on ∇ m − 1 ’ ˛ @ Ω . So ˆ ˆ ∇ m − 1 ’ · ˙ ∇ m ’ · A ∇ m u = M A Ω u dff Ω @ Ω for some ˙ M A Ω u . I f m = 1 then M A Ω u = � · A ∇ u . A free boundary corresponds to ˙ M A Ω u = 0. Ariel Barton The higher-order Neumann problem May 30, 2018 7 / 39
Higher order boundary value problems We are interested in the Dirichlet problems 8 ∇ m · A ∇ m u = 0 in Ω ; > > < ˛ ∇ m − 1 u @ Ω = ˙ ˛ f ; > > : � e N ( ∇ m − 1 u ) � L p ( @ Ω) . � ˙ f � L p ( @ Ω) ; and the Neumann problems 8 ∇ m · A ∇ m u = 0 in Ω ; > > < ˙ M A Ω u = ˙ g; > > : � e N ( ∇ m u ) � L p ( @ Ω) . � ˙ g � L p ( @ Ω) : Nu ( X ) = sup {| u ( Y ) | : | X − Y | < (1 + a ) dist( Y; @ Ω) } Ariel Barton The higher-order Neumann problem May 30, 2018 8 / 39
Higher order boundary value problems We are interested in the Dirichlet problems 8 8 ∇ m · A ∇ m u = 0 in Ω ; ∇ m · A ∇ m u = 0 in Ω ; > > > > < < ˛ ˛ ∇ m − 1 u @ Ω = ˙ ∇ m − 1 u @ Ω = ˙ ˛ ˛ f ; f ; > > > > : : � e N ( ∇ m − 1 u ) � L p ( @ Ω) . � ˙ � e N ( ∇ m u ) � L p ( @ Ω) . �∇ fi ˙ f � L p ( @ Ω) ; f � L p ( @ Ω) ; and the Neumann problems 8 8 ∇ m · A ∇ m u = 0 in Ω ; ∇ m · A ∇ m u = 0 in Ω ; > > > > > < < M A ˙ ˙ M A Ω u = ˙ g; Ω u = ˙ g; > > > > > : � e N ( ∇ m − 1 u ) � L p ( @ Ω) . � ˙ � e N ( ∇ m u ) � L p ( @ Ω) . � ˙ : g � ˙ − 1 ( @ Ω) ; g � L p ( @ Ω) : W p Nu ( X ) = sup {| u ( Y ) | : | X − Y | < (1 + a ) dist( Y; @ Ω) } Ariel Barton The higher-order Neumann problem May 30, 2018 8 / 39
Regularity of coefficients (Caffarelli, Fabes, Kenig, 1981) There is a real, symmetric matrix A , continuous in B ⊂ R 2 , such that ∇ · A ∇ u = 0 in B; u = f on @B; � Nu � L p ( @B ) . � f � L p ( @B ) is ill-posed for all 1 < p < ∞ . Ariel Barton The higher-order Neumann problem May 30, 2018 9 / 39
Regularity of coefficients (Caffarelli, Fabes, Kenig, 1981) There is a real, symmetric matrix e A , continuous in B ⊂ R 2 , such that ∇ · e � · e A ∇ u = 0 in B; A ∇ u = g on @B; � N ( ∇ u ) � L p ( @B ) . � g � L p ( @B ) is ill-posed for all 1 < p < ∞ . Ariel Barton The higher-order Neumann problem May 30, 2018 9 / 39
Regularity of coefficients (Caffarelli, Fabes, Kenig, 1981) There is a real, symmetric matrix e A , continuous in B ⊂ R 2 , such that ∇ · e � · e A ∇ u = 0 in B; A ∇ u = g on @B; � N ( ∇ u ) � L p ( @B ) . � g � L p ( @B ) is ill-posed for all 1 < p < ∞ . ( x; t ) �→ ( x; t − ( x )) u ˜ u I f ∆ u = 0, then ∇ · A ∇ ˜ u = 0, where ! I ∇ ( x ) A ( x; t ) = ∇ ( x ) T 1 + |∇ ( x ) | 2 Notice A ( x; t ) is real, symmetric, and t -independent. Ariel Barton The higher-order Neumann problem May 30, 2018 9 / 39
t -independence and Lipschitz domains From now on we will work with equations of the form X ∇ m · A ∇ m u = @ ¸ ( A ¸˛ @ ˛ u ) = 0 | ¸ | = | ˛ | = m where the coefficient matrix A is elliptic and t-independent, that is, for all x ∈ R d − 1 and all s , t ∈ R . A ( x; t ) = A ( x; s ) = A ( x ) Ariel Barton The higher-order Neumann problem May 30, 2018 10 / 39
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