## New lower semicontinuity results for polyconvex integrals.(English)Zbl 0810.49014

The paper deals with integral functionals of the form $$F(u,\Omega)= \int_ \Omega f(\nabla u)dx$$ initially defined for $$u\in C^ 1(\Omega; \mathbb{R}^ k)$$, $$\Omega\subseteq \mathbb{R}^ n$$. The function $$f$$ is assumed to be polyconvex in the sense of Ball, i.e., there exists a convex function $$g$$ defined in the space of all $$n$$-vectors of $$\mathbb{R}^ n\times \mathbb{R}^ k$$, such that $$f(A)= g({\mathcal M}(A))$$ for every $$k\times n$$ matrix $$A$$, where $${\mathcal M}(A)$$ is the $$n$$-vector whose components are the determinants of all minors of the matrix $$A$$. The aim of the paper is to study the $$L^ 1(\Omega; \mathbb{R}^ k)$$-lower semicontinuous extension $$\mathcal F$$ for $$F$$ defined as the greatest lower semicontinuous functional on $$L^ 1(\Omega; \mathbb{R}^ k)$$ which is less or equal to $$F$$, where $$F(u,\Omega) \triangleq +\infty\;\forall u\in L^ 1(\Omega; \mathbb{R}^ k)\backslash C^ 1(\Omega; \mathbb{R}^ k)$$. It is shown that, if the polyconvex function $$f$$ satisfies the inequality $c_ 0| {\mathcal M}(A)|\leq f(A)\leq c_ 1(|{\mathcal M}(A)|+1)$ with $$0< c_ 0\leq c_ 1$$, then for every function $$u\in \text{BV}(\Omega; \mathbb{R}^ k)$$ with $$\int_ \Omega| {\mathcal M}(\nabla u)| dx< \infty$$ the integral representation ${\mathcal F}(u,\Omega)= \int_ \Omega f(\nabla u) dx$ holds. At the same time an example is given which shows that when $$f(A)= | {\mathcal M}(A)|$$ and $$k\geq 2$$, the set function $$\Omega\to {\mathcal F}(u,\Omega)$$ is not subadditive if $$u$$ is an arbitrary function of $$\text{BV}_{\text{loc}}(\mathbb{R}^ n; \mathbb{R}^ k)$$ so that the functional $$u\to {\mathcal F}(u,\Omega)$$ cannot be represented by an integral over $$\Omega$$. Moreover, a counterexample to the subadditivity of $$\Omega\to {\mathcal F}(u,\Omega)$$ is constructed where $$u\in W^{1,p}_{\text{loc}}(\mathbb{R}^ n; \mathbb{R}^ k)$$ and $$p< \min\{n,k\}$$.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation
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### References:

 [1] Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1977) 337-403 · Zbl 0368.73040 [2] Ball J.M., Murat F.:W 1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225-253, and66 (1986) 439-439 · Zbl 0549.46019 [3] Brezis H., Coron J.M., Lieb E.H.: Harmonic maps with defects. Comm. Math. Phys.107 (1986) 679-705 · Zbl 0608.58016 [4] Buttazzo G.: Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Pitman Res. Notes in Math., Longman, Harlow, 1989 · Zbl 0669.49005 [5] Dacorogna B.: Direct Methods in the Calculus of Variations. Springer, Berlin Heidelberg New York, 1989 · Zbl 0703.49001 [6] De Giorgi E.: Riflessioni su alcuni problemi variazionali. Equazioni Differenziali e Calcolo delle Variazioni (Pisa, 1991) [7] De Giorgi E.: On the relaxation of functionals defined on cartesian manifolds, 33-38, Plenum Press, New York, 1992. Developments in Partial Differential Equations and Applications to Mathematical Physics (Ferrara, 1991) · Zbl 0895.49019 [8] Ekeland I., Temam R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam, 1976 · Zbl 0322.90046 [9] Federer H.: Geometric Measure Theory. Springer, Berlin Heidelberg New York, 1969 · Zbl 0176.00801 [10] Giaquinta M., Modica G., Sou?ek J.: Cartesian currents, weak diffeomorphisms and nonlinear elasticity. Arch. Rational Mech. Anal.106 (1989) 97-159 · Zbl 0677.73014 [11] Giaquinta M., Modica G., Sou?ek J.: Cartesian currents and variational problems for mappings into spheres. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)16 (1989) 393-485 · Zbl 0713.49014 [12] Giaquinta M., Modica G., Sou?ek J.: Erratum and addendum to ?Cartesian currents, weak diffeomorphisms and nonlinear elasticity?. Arch. Rational Mech. Anal.109 (1990) 385-392 · Zbl 0712.73009 [13] Giaquinta M., Modica G., Sou?ek J.: Graphs of finite mass which cannot be approximated in area by smooth graphs. Manus. Math.78 (1993) 259-271 · Zbl 0796.58006 [14] Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston Basel, 1984 · Zbl 0545.49018 [15] Goffman C., Serrin J.: Sublinear functions of measures and variational integrals. Duke Math. J.31 (1964) 159-178 · Zbl 0123.09804 [16] Lebesgue H.: Integrale, longeur, aire. Ann. Mat. Pura Appl.7 (1902) 231-359 · JFM 33.0307.02 [17] Luckhaus S., Modica L.: The Gibbs-Thompson relation within the gradient theory of phase transitions. Archive Rational Mech. Anal.107 (1989) 71-83 · Zbl 0681.49012 [18] Müller S.: Weak continuity of determinants and nonlinear elasticity. C. R. Acad. Sci. Paris Sèr. I Math.307 (1988) 501-506 · Zbl 0679.34051 [19] Meyers N.G., Serrin J.: H=W. Proc. Nat. Acad. Sci. U.S.A.51 (1964) 1055-1056 · Zbl 0123.30501 [20] Miranda M.: Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)18 (1964) 515-542 · Zbl 0152.24402 [21] Morrey C.B. Jr.: Quasi-convexity and the semicontinuity of multiple integrals. Pacific J. Math.2 (1952) 25-53 · Zbl 0046.10803 [22] Morrey C.B. Jr.: Multiple Integrals in the Calculus of Variations. Springer, Berlin Heidelberg New York, 1966 · Zbl 0142.38701 [23] Reshetnyak Yu.G.: Weak convergence of completely additive vector functions on a set. Siberian Math. J.9 (1968) 1039-1045 · Zbl 0176.44402 [24] Serrin J.: On the definition and properties of certain variational integrals. Trans. Am. Math. Soc.101 (1961) 139-167 · Zbl 0102.04601 [25] Simon L.M.: Lectures on Geometric Measure Theory. Proc. of the Centre for Mathematical Analysis (Canberra), Australian National University, 3, 1983 · Zbl 0546.49019 [26] Vainberg M.M.: Variational Methods for the Study of Non-Linear Operators. Holden-Day, San Francisco, 1964
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