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Consider a uniformly elliptic equation $$ \sum_{i,j=1}^n a_{ij}(x)\partial_{ij}u+\sum_{i=1}^n b_{i}(x)\partial_{i}u+c(x)u=0 $$ say, in an open ball $B\subset \mathbb R^n$, where coefficients are Hölder continuous in $\bar B$ with some exponent $\alpha\in(0,1).$ There are interior Schauder estimates, of course. But they are conditional, supposing that a solution is from $C^{2+\alpha}$ locally then giving an estimate for it.

Is somewhere stated that all solutions of this equation has to be classical in $B$, i.e. from $C^2(B)$? Or can it be derived from some other considerations? They should be even from $C^{2+\alpha}(B)$, but I can't find a reference.

The same question goes for solutions of a uniformly parabolic equation $$ \partial_tu-\sum_{i,j=1}^n a_{ij}(x,t)\partial_{ij}u-\sum_{i}^n b_{i}(x,t)\partial_{i}u-c(x,t)u=0 $$ with Hölder continuous coefficients.

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What kind of solutions are you considering? –  Mike Hall May 31 '12 at 8:29
    
@Mike Hall strong solutions: twice weakly differentiable and satisfying the equation a.e. If there is some other suitable class of solutions, the answer would be interesting too. –  Andrew May 31 '12 at 10:26
    
Yeah, unfortunately all I really know to do is go through the appropriate chapters of Gilbarg and Trudinger very carefully. They have some pretty complete results for weak solutions, with the operator in divergence form, in Chapter 8, and then in Chapter 9 do some $L^p$ theory for strong solutions. Just skimming through, it looks like the $L^p$ estimates they give might only give you $C^{1,\alpha}$ for some $\alpha$, but it's hard to tell from just jumping into the middle. –  Mike Hall Jun 5 '12 at 9:39
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2 Answers

The Schauder estimates are what you need. They are a priori estimates, but they're the estimates you feed into the method of continuity, which guarantees the $C^{2,\alpha}$ interior regularity of your solution. See, for example, section 6.3 ("The Dirichlet Problem") in Gilbarg and Trudinger.

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You can start with weak solutions in $H^1$ and use the $L^2$-regularity theory to get $H^s$-type smoothness, which then would guarantee classical derivatives by the Sobolev embedding. This approach can be learned from practically any textbook on PDE. Examples are Folland's Introduction to PDE, Evans' PDE, and Jost's PDE.

Another way is to use a Campanato space approach to get Hölder continuity of solutions, and then use the Schauder theory as you suggested. This can be read in Giaquinta's Multiple integrals in the calculus of variations, Han and Lin's Elliptic PDE, and Chen and Wu's Second order elliptic equations and elliptic systems. This approach has an advantage that it can be used as a stepping stone to the De Giorgi-Nash-Moser regularity theory.

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@timur as far as I understand all what is for equations in divergence form. –  Andrew Jun 19 '12 at 18:23
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