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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
##### Citations:
Zbl 0007.11804; Zbl 0494.49031
Full Text:
##### 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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