Boundary regularity of solutions of degenerate elliptic equations without boundary conditions

Transcription

1 Boundary regularity of solutions of elliptic without boundary Iowa State University November 15, 2011

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3 If the linear second order elliptic operator L is non with smooth coefficients, then, for any smooth boundary values ϕ on a smooth domain Ω R n, there is a smooth solution of the Dirichlet assuming that L1 0. Lu = f in Ω, u = ϕ on Ω,

4 If L is, however, then this may not be solvable for arbitrary smooth ϕ. Fichera developed a general theory of such s which was further extended by Oleinik and Radkevich. In particular, no boundary condition need be prescribed on certain parts of the boundary, determined by the behavior of the coefficients of L near an individual point on the boundary.

5 Here, we analyze a special situation in more detail. Write the operator as Lu = a ij D ij u + b i D i u + cu and suppose that [a ij ] is a uniformly positive definite matrix, but the vector b satisfies the condition that b d, where d is distance to Ω, is bounded from below by a positive constant near Ω.

6 In this case, we write b i = β i /d and γ = c/d, and we assume that β Dd > 0 on the boundary. We consider the d u + β i D i u + γu = ϕ in Ω. We also assume that β is bounded. (More general highest order terms may be considered.

7 Langlais showed in 1984 that this has a unique solution for each smooth ϕ and that the is smooth assuming that γ is bounded from above by a sufficiently large negative constant. His method was to estimate solutions of the approximating boundary value (d + ε)a ij D ij u + β i D i u + γu = ϕ in Ω, β i D i u + γu = ϕ on Ω with a parameter ε 0.

8 In fact, he only proved a bound on the second s of u and then quoted a general theorem which requires some heavy machinery due to Bolley and Camus. This machinery requires β Dd > 1 2 near Ω.

9 For other reasons, I was looking at the oblique u = f in Ω, β i D i u + γu = ϕ on Ω, with f = O(d α 1 ) for some α (0, 1), and I observed two things: v = β i D i u + γu ϕ (with β, γ, ϕ extended into Ω) solves a Dirichlet, so one can estimate the Hölder norm v α rather easily. The estimate for v gives a Hölder estimate for Du.

10 For our, by looking locally, we may assume that β and ϕ is constant and γ 0, and then v satisfies the (d + ε) v + β i D i v β Dd v = 0 in Ω, d + ε v = 0 on Ω. It s not difficult to adapt previous arguments to get a uniform Hölder estimate for v, so v = O(d α ) and then use the equation u = v d + ε to conclude that u = O(d α 1 ).

11 In fact, we can get Schauder type estimates, so our original has a solution in C 1,α 0 for some α 0 (0, 1) (determined by Ω) if Ω is a Lipschitz domain, and the solution is in C k,α if the coefficients a ij are in C k 2,α, β and γ are in C k 1,α and Ω C k 1,α for any given α (0, 1).

12 These ideas also show how the elliptic equation is like the solution of the ordinary differential equation u + a x u = 0 near x = 0. If a 1, the general solution is u = c 1 + c 2 x 1 a so all solutions are globally continuous at x = 0 if a < 1. If a > 0, the only globally C 1 solutions are constants, and if a > 1, then the only bounded solutions are constants. The constant solutions are the unique solutions of the ODE with Neumann condition u (0) = 0.

13 For the higher-dimensional (that is, PDE) case, we have a similar situation but nonzero Neumann-like data. This method shows that there is a unique C 1 solution if β Dd > 0 on Ω and it s as smooth as data allow. If β Dd > 1, then there is a unique bounded solution of the differential equation, and it s as smooth as data allow. We also don t need any restrictions on γ (except for being negative).