Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^q(\Omega), \ \ q>n/2.$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})\cap W^{2,1}{(B)}$?
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1$\begingroup$ While "two times differentiable almost everywhere" is somewhat ambiguous, the statement is trivially false as written regardless of interpretation (take e.g. $u$ a primitive of a Cantor function in 1D, $f = 0$). Perhaps you meant to put some further assumption on $u$? $\endgroup$– user378654Commented Feb 10, 2023 at 5:30
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1$\begingroup$ user378654- I edited the question! $\endgroup$– DejvCommented Feb 10, 2023 at 7:34
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