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Convex functionals and partial regularity. (English) Zbl 0658.49005
The authors study the regularity properties of local minimizers u: \({\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\) of the functional \[ F(u,\Omega):=\int_{\Omega}f(\cdot,u,Du)dx \] with integrand f(x,y,P) convex in P and of growth order \(m\geq 1\), i.e. \[ c_ 1| P|^ m\leq f(x,y,P)\leq c_ 2\cdot (1+| P|^ m). \] For exponents \(m>1\) the minimizer is in the Sobolev space \(H^{1,m}_{loc}(\Omega,{\mathbb{R}}^ N)\), for \(m=1\) the problem is discussed in \(BV_{loc}(\Omega,{\mathbb{R}}^ N)\). In contrast to the known results no global assumptions concerning the regularity and ellipticity of the integrand are imposed, the main regularity theorem only involves a local criterion which in case of integrands \(f=f(P)\) can be summarized as follows: If u is a local F-minimizer with convex integrand f: \({\mathbb{R}}^{nN}\to {\mathbb{R}}\) and if for some points \(x_ 0\in \Omega\), \(\bar P\in {\mathbb{R}}^{nN}\) \[ \lim_{R\downarrow 0}\int_{B_ R(x_ 0)}| Du-\bar P|^ m=0 \] holds, then u is of class \(C^{1,\alpha}\) near \(x_ 0\), provided f is smooth in a neighborhood of \(\bar P\) and satisfies \(f_{p^ i_{\alpha}p^ j_{\beta}}(\bar P) Q^ i_{\alpha} Q^ j_{\beta}\leq \lambda | Q|^ 2\) with \(\lambda >0\). The proof is based on decay estimates for a quantity which measures the deviation of Du from being Hölder continuous. In order to get these inequalities one has to compare the solution u with minimizers of frozen functionals, another key ingredient entering the proof is a smoothing lemma stated in section 4 of the paper. It is worth noting that the cases \(m=1\) and \(m>1\) can be handled simultaneously with the above described unified approach.
Reviewer: M.Fuchs

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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[1] E. Acerbi & N. Fusco: A regularity theorem for minimizers of quasiconvex integrals, Preprint 1986. · Zbl 0627.49007
[2] F. J. Almgren, Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. 87 (1968) 321–391. · Zbl 0162.24703 · doi:10.2307/1970587
[3] G. Anzellotti: Parametric and non-parametric minima, Manuscripta Math. 48 (1984), 103–115. · Zbl 0564.49026 · doi:10.1007/BF01169003
[4] G. Anzellotti & M. Giaquinta: Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova 60 (1978) 1–21. · Zbl 0432.46031
[5] E. Bombieri: Regularity theory for almost minimal currents, Arch. Rational Mech. Anal. 78 (1982) 99–130. · Zbl 0485.49024 · doi:10.1007/BF00250836
[6] E. de Giorgi: Frontiere orientate di misura minima. Sem. Mat. Scuola Normale Superiore Pisa, 1961. · Zbl 0296.49031
[7] L. C. Evans: Quasiconvexity and partial regularity in the calculus of variations, Preprint 1984.
[8] M. Giaquinta: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Math. Studies n105, Princeton University Press, Princeton 1983. · Zbl 0516.49003
[9] M. Giaquinta: The problem of the regularity of minimizers. Address at the international Congress of Mathematicians, Berkeley 1986. · Zbl 0607.49003
[10] M. Giaquinta: Quasiconvexity, growth conditions and partial regularity. Preprint 1987. · Zbl 0638.49005
[11] M. Giaquinta & E. Giusti: On the regularity of minima of variational integrals. Acta Math. 148 (1982) 31–46. · Zbl 0494.49031 · doi:10.1007/BF02392725
[12] M. Giaquinta & E. Giusti: Quasi-minima. Ann. Inst. H. Poincaré. Analyse non linéaire 1 (1984) 79–107. · Zbl 0541.49008
[13] M. Giaquinta & G. Modica: Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse non linéaire 3 (1986) 185–208.
[14] M. Giaquinta & G. Modica: Remarks on the regularity of the minimizers of certain degenerate functional. Manuscripta Math. 57 (1986) 55–99. · Zbl 0607.49003 · doi:10.1007/BF01172492
[15] M. Giaquinta, G. Modica, & J. Souček: Functional with linear growth in the calculus of variations. Comment. Math. Univ. Carolinae 20 (1979) 143–172. · Zbl 0409.49007
[16] E. Giusti: Boundary value problems for nonparametric surfaces of prescribed mean curvature. Ann. Sc. Norm. Sup. Pisa 3 (1976) 501–541. · Zbl 0344.35036
[17] M. Miranda: Analiticitá delle superfici di area minima in \(\mathbb{R}\)4. Rend. Acc. Naz. Lincei (1965). · Zbl 0135.32601
[18] R. E. Reifenberg: On the analyticity of minimal surfaces. Annals of Math. 80 (1964) 15–21. · Zbl 0151.16702 · doi:10.2307/1970489
[19] Yu. G. Reschetnyak: Weak convergence of completely additive vector functions on a set. Sibirsk. Maz. Ž. 9 (1968) 1386–1394 (Translated).
[20] R. Schoen & L. Simon: A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals. Indiana Univ. Math. J. 31 (1982) 415–434. · Zbl 0516.49026 · doi:10.1512/iumj.1982.31.31035
[21] R. Schoen & K. Uhlenbeck: A regularity theory for harmonic maps. J. Diff. Geo. 17 (1982) 307–335. · Zbl 0521.58021
[22] L. Tamanini: Regularity results for almost minimal oriented hypersurfaces in \(\mathbb{R}\)m. Quaderni Dip. Mat. Univ. Lecce, n1 (1984). · Zbl 1191.35007
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