zbMATH — the first resource for mathematics

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.

MSC:
 49N60 Regularity of solutions in optimal control
Zbl 0802.35016
Full Text:
References:
  Giaquinta, J. Reine Angev. Math 311/312 pp 145– (1979)  DOI: 10.1007/BF02392725 · Zbl 0494.49031  Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems 105 (1983) · Zbl 0516.49003  DOI: 10.1007/BF01234312 · Zbl 0791.49041  DOI: 10.2307/2946602 · Zbl 0785.30009  DOI: 10.1215/S0012-7094-75-04211-8 · Zbl 0347.35039  DOI: 10.1080/03605309308820984 · Zbl 0796.35061  DOI: 10.1515/crll.1994.454.143 · Zbl 0802.35016  Meyers, Ann. Scuola Norm. Sup. Pisa 17 pp 189– (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.