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Growth conditions and regularity. A counterexample. (English) Zbl 0638.49005
Suppose that \(F: {\mathbb{R}}\) \(n\to {\mathbb{R}}\) is a strictly convex function. Then it is well-known [T. Radò, On the problem of Plateau (1933; Zbl 0007.11804)] that the Dirichlet-problem associated to the variational integral \(\int F(Du)dx\) and for smooth boundary data has a unique solution in the class of Lipschitz-functions. On the other hand if F is not convex but of growth \(m>2\), i.e. \[ (1)\quad c_ 0(| p| \quad m-1)\leq F(p)\leq c_ 1(| p| \quad m+1), \] then the techniques of M. Giaquinta and E. Giusti [Acta Math. 148, 31- 46 (1982; Zbl 0494.49031)] show that local minimizers in the space \(H^{1,m}\) are locally bounded and Hölder continuous. In the present note the author constructs examples of unbounded local minimizers with F strictly convex but satisfying a growth condition of the form \[ c_ 0(| p|^ 2-1)\leq F(p)\leq c_ 1(| p| \quad m+1) \] for some exponent \(m>2\). Thus (1) is a necessary condition for a local regularity.
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
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References:
[1] M. Giaquinta, E. Giusti: On the regularity of minima of variational integrals. Acta Math.148 (1982), 31-46 · Zbl 0494.49031
[2] M. Miranda: Un teorema di esistenza e unicità per il problema dell’area minima in n variabili. Ann. Sc. Norm. Sup. Pisa19 (1965), 233-249 · Zbl 0137.08201
[3] T. Radò: On the problem of Plateau. Springer-Verlag, 1933 · JFM 59.1341.01
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