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On the regularity of very weak minima. (English) Zbl 0851.49026
Consider a domain \(\Omega \subset \mathbb{R}^n\) and a function \(A : \Omega \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R}_{nN}\) satisfying \[ \begin{aligned} & A(x,y,Q) \cdot Q \geq a |Q |^q, \\ & \bigl |A (x,y,Q_1) - A(x,y,Q_2) \bigr |\leq b |Q_1 - Q_2 |\bigl( |Q_1 |^{p - 2} + |Q_2 |^{p - 2} \bigr), \\ & \bigl |A(x,y,0)\bigr |\leq h(x) + d |y |^{p - 1} \end{aligned} \] for almost every \(x \in \Omega\), for every \(y \in \mathbb{R}^N\) and all \(Q, Q_1,Q_2 \in \mathbb{R}^{nN}\). Here \(a,b > 0\), \(d \geq 0\) and \(p \geq 2\) denote constants, \(h \in L^{p/(p-1)} (\Omega)\) is a nonnegative function. The authors prove the existence of a number \(r_1 \in (p - 1,p)\) depending only on the data with the following property: if \(u \in W^{1, r_1}_{ \text{loc}} (\Omega, \mathbb{R}^N)\) satisfies \[ \int_\Omega A(x,u, \nabla u) \cdot \nabla \varphi dx = 0 \quad \forall \varphi \in C^1_0 (\Omega, \mathbb{R}^N), \] then \(u\) is in the space \(W^{1,p}_{\text{loc}} (\Omega, \mathbb{R}^N)\). A similar result has been obtained by T. Iwaniec and C. Sbordone [J. Reine Angew. Math. 454, 143-161 (1994; Zbl 0802.35016)] under the additional assumption that \(A\) is homogeneous w.r.t. the last argument.

49N60 Regularity of solutions in optimal control
Zbl 0802.35016
Full Text: DOI
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