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Régularité de la solution d’un problème variationnel. (Regularity of the solution of a variational problem). (French) Zbl 0757.49015

The author proves an existence result for the following problem \[ \text{Inf}\left\{\int_ \Omega[g(x,\nabla v)+l(x,v)]dx:v\in\psi+W_ 0^{1,p}(\Omega)\right\}, \tag{\(P_ 1\)} \] \(g\) being a function verifying suitable assumptions but not necessarily convex in the gradient variables and \(p\geq 2\). The basic idea to do this is to consider the relaxed problem \((P_ 1R)\) of \((P_ 1)\), namely \[ \text{Min}\left\{\int_ \Omega[g^{**}(x,\nabla v)+l(x,v)]dx:v\in\psi+W_ 0^{1,p}(\Omega)\right\}, \tag{\(P_ 1R\)} \] where for every \(x\) in \(\Omega\), \(g^{**}(x,\cdot)\) denotes the convex envelope of \(g(x,\cdot)\), to observe that \((P_ 1R)\) has in general a solution, say \(u\), and to prove that the set \(E=\{x\in\Omega:g^{**}(x,\nabla u(x))\leq g(x,\nabla u(x))\}\) has zero measure. In order to perform the above described proof it is assumed that the function \(g^{**}\) is affine in the second set of variables on each connected component of the set \(\{(x,z)\in\Omega\times\mathbb{R}^ n\): \(g^{**}(x,z)\leq g(x,z)\}\). Moreover, in order to prove that the set \(E\) has zero measure, it is observed that a regularity result for the solutions of a variational problem is needed. More precisely if \(l(x,\cdot)\) is convex a dual problem of \((P_ 1)\) must be considered and if \(l(x,\cdot)\) is not convex the regularity of the solution \(\overline p=(\overline p_ 1,\dots,\overline p_ n)\) of the problem \[ \overline p_ i={\partial g^{**}\over\partial z_ i}(x,\nabla u),\quad i=1,2,\dots,n;\quad\text{div }\overline p={\partial l\over\partial v}(x,u) \tag{E} \] is needed. By virtue of this the regularity of the solutions of the following problem, called the dual problem of \((P_ 1R)\), is first studied: \[ \text{Inf}\left\{\int_ \Omega[g^*(x,-w)+l^*(x,-\text{div} w)]dx: w\in(L^ q(\Omega))^ n, \text{ div} w\in L^ q(\Omega)\right\}, \tag{\(P_ 2\)} \] where \(g^*(x,\cdot)\) and \(l^*(x,\cdot)\) are the polars of \(g(x,\cdot)\) and \(l(x,\cdot)\) and \(1/p+1/q=1\). It is proved that the solution \(\overline p\) of \((P_ 2)\) exists, that it is in \((W^{1,q}_{loc}(\Omega))^ n\), that \(\text{div} \overline p\in W^{1,q}_{loc}(\Omega)\) and that \((E)\) holds. Building on this result it is also proved that \(u\) is in \(W^{l,np/(n-q)}_{loc}(\Omega)\cap L^{np/(n-(p+q))}_{loc}(\Omega)\) and the existence of the solution of \((P_ 1)\) is obtained. Analogous results are also proved for problems involving integral functionals of the type \(\int_ \Omega g(x,u,\nabla u)\).
Finally some examples verifying the assumptions of the above results are discussed and some sufficient conditions in order to achieve these assumptions are given.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
58J32 Boundary value problems on manifolds
74A20 Theory of constitutive functions in solid mechanics
58E30 Variational principles in infinite-dimensional spaces
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References:

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