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I know that there are a lot of (great) books treating regularity of weak solutions of elliptic pdes (such as Gilbarg-Trudinger), but what about regularity of very weak solutions, that is, solutions in the distributional sense? For concreteness, consider a bounded domain $\Omega \subset \mathbb{R}^2$ with smooth boundary, and two continuous functions $u, f: \Omega \to \mathbb{R}$ satisfying $-\Delta u = f$ on $\Omega$ in the distributional sense, that is $$ -\int_{\Omega}{u\Delta \phi \mathrm{d}x} = \int_{\Omega}{f \phi\mathrm{d}x}\quad \forall \phi \in \mathscr{C}^{\infty}_{\text{c}}(\Omega). $$ I've read in many articles statements like "if $u \in L^{\infty}_{\text{loc}}(\Omega)$, then by standard elliptic regularity $u \in \mathscr{C}^{1, \alpha}_{\text{loc}}(\Omega)$". What do they mean with standard elliptic regularity in this case? Any help would be appreciated, even just a reference.

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Just a remark: weak solutions are solutions the have weak derivatives. In other words, the distributional derivatives must be (square) integrable. –  Siminore Aug 17 '12 at 10:01
    
Uhm, I think this should read "if $f\in L^\infty_{loc}$, then". In case this is true this is basically covered by Gilbarg Trudinger, Theorem 8.34. –  user20266 Aug 20 '12 at 7:34
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Was it a bounty question? First time in my life I got awarded 500 points! –  timur Aug 23 '12 at 4:46
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3 Answers

The term "Standard elliptic theory" loosely designates the set of results obtained mainly during 1950-1960's on linear elliptic equations. Sometimes it is mentioned in the context of nonlinear equations, but it would usually mean that the result is a straightforward consequence of the "linear standard elliptic theory". The theory has two popular versions, that respectively take place in Hölder spaces and in Sobolev spaces. In general it can be described by the following scheme.

Suppose that $\Omega$ is a bounded domain, $A$ is a linear elliptic operator on $\Omega$, and $\{B_j:j=1,\ldots,n\}$ are collection of linear operators on the boundary. Then we consider the boundary value problem $Au=f$ and $B_ju=g_j$, ($j=1,\ldots,n$).

To proceed, one has to choose:

  • Scale $X^s$ of functions spaces on $\Omega$, and
  • Scale $Y^s$ of functions spaces (typically of the form $Y=Y^s_1\times\cdots\times Y_n^s)$ on $\partial\Omega$, satisfying $$ A:X^s\to X^{s-2m},\qquad\textrm{and}\qquad B_j:X^s\to Y_j^s,\qquad\textrm{are bounded.} $$

Then one proves (for some range of $s$):

  • Elliptic estimates: $$ \|u\|_{X^s}\lesssim\|{Au}\|_{X^{s-2m}}+\|{B_1}\|_{Y_1^s}+\cdots+\|{B_mu}\|_{Y^s_n}+\|{u}\|_{X^0}. $$
  • Elliptic regularity: $$ Au\in X^{s-2m},\ \{B_ju\}\in Y^s\iff u\in X^s. $$
  • Fredholm property: $$ u\mapsto (Au,\{B_ju\}):X^s\to X^{s-2m}\times Y\qquad \textrm{is Fredholm}. $$ i.e, its kernel and cokernel are finite dimensional.

Here $2m$ is usually the order of the operator, but if you want to go beyond usual elliptic operators sometimes one has suboptimal results. Of course, the theory itself has to decide what operators $A$ it can include, and what compatibility conditions have to be imposed on the operators $A$ and $\{B_j\}$. Also, if coefficients of $A$ and $B_j$ are not smooth, the theory will be valid for some limited range of $s$.

A prototypical example is Schauder's theory for second order elliptic equations in Hölder spaces.

Such a theory for very general elliptic systems in Hölder and Sobolev type spaces was established in 50'-60's. cf. Agmon-Douglis-Nirenberg 1959, and 1964.

There is also a version of the theory for strongly elliptic systems, that assumes a bit more but delivers stronger results. This was worked out by Garding, Nirenberg, Agmon, et al.

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Here is an example of elliptic regularity theory for distributional solutions.

Theorem. Suppose $\Omega$ is on open set in $\mathbb{R}^n$ and $L$ is an elliptic operator of order $k$ with $C^\infty$ coefficients on $\Omega$. Let $u$ and $f$ be distributions on $\Omega$ such that $Lu=f$. If $f \in H_s^{\text{loc}}(\Omega)$ or some $s \in \mathbb{R}$, then $u \in H_{s+k}^{\text{loc}}(\Omega)$.

This is proved as Theorem 6.33 of the book by Gerald B. Folland, Introduction to partial differential equations, Princeton University Press, second edition.

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An important theorem in this field is Weyl's Lemma:

Lemma (Weyl): If for some $u\in L^1_{\text{loc}}(\Omega)$ and all test-functions $v$ it holds that $$\int_\Omega u(x)\Delta v(x)dx = 0,$$ then $u$ is $C^\infty(\Omega)$.

A proof can be found in Zeidler's "Nonlinear Functional Analysis, II/A - Linear Monotone Operators" Theorem 18.G.

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