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Convex Functions Methods in the Dirichlet Problem for Euler—Lagrange Equations

In: Variational Methods for Free Surface Interfaces

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  • Ilya J. Bakelman

Abstract

In this paper we investigate a priori estimates for solutions of the second order elliptic E—L equations,* whose gradients satisfy some prescribed limitations. Such problems arise from the relativity theory and continuous mechanics and can be described in terms of variational problems for the n-dimensional multiple integrals 1 $$\int_{B}\, F(x, u, Du)\, dx$$ whose integrands F(x, u, p) are defined only for vectors p belonging to prescribed domain G in R n . If G coincides with the whole space R n , then we do not have any prescribed limitations for the gradient of desired solutions for the E—L equation corresponding to the functional (1). This most simple case was investigated in our paper [1].

Suggested Citation

  • Ilya J. Bakelman, 1987. "Convex Functions Methods in the Dirichlet Problem for Euler—Lagrange Equations," Springer Books, in: Paul Concus & Robert Finn (ed.), Variational Methods for Free Surface Interfaces, pages 127-137, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-4656-5_15
    DOI: 10.1007/978-1-4612-4656-5_15
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