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Existence and partial regularity in the calculus of variations. (English) Zbl 0648.49008
In the first part of the paper the author considers integrals of the form \[ (1)\quad {\mathcal F}(u(\Omega)=\int_{\Omega}f(\cdot,u,Du)dx \] defined for vector valued functions \(u: {\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\) and with the additional property that the integrand \(f(x,y,Q)\) satisfies the growth condition \(f_ 0(Q)-g(x)\leq f(x,y,Q)\leq \lambda | Q|^ m+g(x)\) with \(\lambda,g>0\), \(m\geq 2\) and \(f_ 0\) quasiconvex with growth order m. He then introduces a concept of generalized quasi-minima \(u\in H^{1,m}(\Omega,{\mathbb{R}}^ N\)) of the functional (1) and shows \(Du\in L^{m+\epsilon}_{loc}(\Omega,{\mathbb{R}}^{nN})\) for some small \(\epsilon >0\). The proof uses a version of Caccioppoli’s inequality being valid for generalized quasi-minima.
A second chapter gives some existence theorems for minimizers of quasiconvex functionals (1), a final section is concerned with the partial regularity properties of these minimizers in a special case. Related results can be found for example in a paper by M. Giaquinta and G. Modica [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 185-208 (1986; Zbl 0594.49004)].
Reviewer: M.Fuchs

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
49J20 Existence theories for optimal control problems involving partial differential equations
26B25 Convexity of real functions of several variables, generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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