On the lower semicontinuity of quasiconvex integrals in \(\text{SBV}(\Omega ,\mathbf R^ k)\).

*(English)*Zbl 0817.49017This paper is concerned with a certain class of functionals which arise in the calculus of variations. Such functionals are typically of the form
\[
F(u)= \int_ \Omega f(x, u,\nabla u) dx+ \int_{S_ u} \phi(u^ +, u^ -, n_ u) d{\mathcal H}^{n- 1}
\]
in which \(\Omega\) is an open set in \(\mathbb{R}^ n\), \(f\) is a Carathéodory function, \(S_ u\) is the jump set of \(u\) (the complement of the Lebesgue set of \(u\)), \(n_ u\) is the measure-theoretic normal to \(S_ u\), and \(u^{\pm}\) are the one-sided traces of \(u\) on both sides of \(S_ u\).

A key question is that of the conditions under which such a functional is lower semicontinuous. The resolution of this question has implications for existence studies using the direct method of the calculus of variations. The functional in question is considered in the space \(\text{SBV}(\Omega, \mathbb{R}^ n)\) of special functions of bounded variation. A function belongs to this space if it is of bounded variation and if, furthermore, \(| D^ s u|\) is supported in \(S_ u\). A study of SBV indicates that the appropriate kind of lower semicontinuity to seek is that with respect to \(L^ 1_{\text{loc}}(\Omega, \mathbb{R}^ n)\), since SBV enjoys a compactness property with respect to this space.

The bulk of the paper is taken up with proving the following result: if \(f(x, s, p)\) is quasiconvex with respect to \(p\) and satisfies a technical growth condition, and if \(\phi\) is a Borel function which is even in its last argument, then the functional \(F\) is lower semicontinuous with respect to \(L^ 1_{\text{loc}}\). This result extends an earlier one, due to the author [Arch. Ration. Mech. Anal. 111, No. 4, 291-322 (1990; Zbl 0711.49064)], in which \(f\) is assumed to be convex in its last argument. The method of proof follows the strategy of E. Acerbi and N. Fusco [Arch. Ration. Mech. Anal. 86, 125-145 (1984; Zbl 0565.49010)].

A key question is that of the conditions under which such a functional is lower semicontinuous. The resolution of this question has implications for existence studies using the direct method of the calculus of variations. The functional in question is considered in the space \(\text{SBV}(\Omega, \mathbb{R}^ n)\) of special functions of bounded variation. A function belongs to this space if it is of bounded variation and if, furthermore, \(| D^ s u|\) is supported in \(S_ u\). A study of SBV indicates that the appropriate kind of lower semicontinuity to seek is that with respect to \(L^ 1_{\text{loc}}(\Omega, \mathbb{R}^ n)\), since SBV enjoys a compactness property with respect to this space.

The bulk of the paper is taken up with proving the following result: if \(f(x, s, p)\) is quasiconvex with respect to \(p\) and satisfies a technical growth condition, and if \(\phi\) is a Borel function which is even in its last argument, then the functional \(F\) is lower semicontinuous with respect to \(L^ 1_{\text{loc}}\). This result extends an earlier one, due to the author [Arch. Ration. Mech. Anal. 111, No. 4, 291-322 (1990; Zbl 0711.49064)], in which \(f\) is assumed to be convex in its last argument. The method of proof follows the strategy of E. Acerbi and N. Fusco [Arch. Ration. Mech. Anal. 86, 125-145 (1984; Zbl 0565.49010)].

Reviewer: B.D.Reddy (Rondebosch)

##### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

49J27 | Existence theories for problems in abstract spaces |

49J40 | Variational inequalities |

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\textit{L. Ambrosio}, Nonlinear Anal., Theory Methods Appl. 23, No. 3, 405--425 (1994; Zbl 0817.49017)

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##### References:

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