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Boundedness of solutions to variational problems under general growth conditions. (English) Zbl 0892.35048

From the author’s introduction: The present paper deals with minimum problems of the calculus of variations and quasilinear elliptic equations in divergence form. The minimum problems we take into account have the form \[ \min\int_GF(x,v,Dv)dx, \quad v=u_0 \text{ on } \partial G. \tag{1} \] Here \(G\) is an open subset of \(\mathbb{R}^n\), whose Lebesgue measure \(m(G)\) is finite; \(n\geq 2\); \(F\) is a Carathéodory function from \(G\times \mathbb{R} \times \mathbb{R}^n\) into \(\mathbb{R}\); \(D\) stands for gradient; \(u_0\) is a prescribed boundary datum. Our assumptions on \(F\) amount to requiring that \(A,B\) and \(s_0\) exist such that \[ F(x,s,\xi) \geq A\bigl( |\xi| \bigr)- B\bigl(| s| \bigr), \quad F(x,s,0)\leq B\bigl(| s|\bigr) \] for \(| s|\geq s_0\), \(\xi\in \mathbb{R}^n\) and a.e. \(x\in G\), where \(|\xi |\) denotes the Euclidean norm of \(\xi\). Here, \(s_0\) is a nonnegative number, \(A\) is a Young function, i.e. a convex increasing function from \([0,\infty)\) into \([0,\infty)\) vanishing at 0, and \(B\) is an increasing function from \([0,\infty)\) into \([0,\infty)\). The boundary datum \(u_0\) is assumed to be a bounded weakly differentiable function on \(\mathbb{R}^n\) such that \(\int A(| Du_0 |) dx< \infty\). The competing functions \(v\) in problem (1) are taken from the class \(K^A_{u_0}\) defined as
\(K^A_{u_0} =\{v| v\) is weakly differentiable in \(G\), \(\int_G A(| Dv |) dx<\infty\) and the continuation of \(v-u_0\) by 0 outside \(G\) is weakly differentiable in \(\mathbb{R}^n\}\).
We are concerned with conditions on \(A\) and \(B\) ensuring that any minimizer of problem (1) is bounded in \(G\).
Reviewer: V.Mustonen (Oulu)

MSC:

35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI

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