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Weak minima of variational integrals. (English) Zbl 0802.35016
Consider the variational equation on $$\Omega \subseteq \mathbb{R}^ n$$ $\text{div} A (x,\nabla u) = 0 \tag{1}$ where $$A:\Omega \times \mathbb{R}^{m \times n} \to \mathbb{R}^{m \times n}$$, $$A(x,\xi) \cong | \xi |^ p$$, $$p>1$$. The definition of a very weak solution of (1) is given as follows. Let $$\max \{p - 1,1\} < r < p$$ and $$u \in W^{1,r}_{\text{loc}} (\Omega, \mathbb{R}^ m)$$ satisfy the integral identity $\int_ \Omega \bigl \langle A(x, \nabla u) |\nabla\varphi \bigr \rangle dx = 0$ for any $$\varphi \in W_ 0^{1,{r/r - p + 1}} (\Omega, \mathbb{R}^ m)$$. Then $$u$$ is called a very weak solution of (1). One of the results proved is the following regularity theorem.
Theorem: There exists $$r_ 1 = r_ 1 (p,u)$$, $$p-1<r_ 1<p$$, such that every very weak solution $$u \in W^{1,r_ 1}_{\text{loc}} (\Omega, \mathbb{R}^ m)$$, actually belongs to $$W^{1,p}_{\text{loc}} (\Omega, \mathbb{R}^ m)$$.
Reviewer: T.Iwaniec

MSC:
 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J60 Nonlinear elliptic equations 35J50 Variational methods for elliptic systems
Keywords:
very weak solution
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