## Regularity for minimizers of non-quadratic functionals: The case $$1<p<2$$.(English)Zbl 0686.49004

Hölder continuity for the gradient of minimizers of the functional $$\int | Du|^ pdx$$, with u: $$\Omega$$ $$\subset {\mathbb{R}}^ n\to {\mathbb{R}}^ N$$, is known in the cases [p$$\geq 2$$, $$N\geq 1]$$ and $$[1<p<2$$, $$N=1]$$, due respectively to K. Uhlenbeck [Acta Math. 138, 219-240 (1977; Zbl 0372.35030)] and E. Di Benedetto [Nonlinear Anal., Theory Methods Appl. 7, 827-850 (1983; Zbl 0539.35027)].
In this paper the authors prove a result of partial $$C^{1,\alpha}$$ regularity for minimizers of the functional $$\int f(x,u(x),| Du(x)|)dx$$ in the case $$1<p<2$$, $$N\geq 1$$, where f(x,s,t) is a convex function of t such that $$c_ 1| t|^ p\leq f(x,s,t)\leq c_ 2| t|^ p$$.
Reviewer: E.Acerbi

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 35D10 Regularity of generalized solutions of PDE (MSC2000)

### Citations:

Zbl 0372.35030; Zbl 0539.35027
Full Text:

### References:

 [1] Di Benedetto, E, C1 + x local regularity of weak solutions of degenerate elliptic equations, Nonlinear anal., 7, 827-850, (1983) · Zbl 0539.35027 [2] {\scN. Fusco and J. Hutchinson}, Partial regularity for minimisers of certain functionals having nonquadratic growth, Ann. Mat. Pura Appl., in press. · Zbl 0698.49001 [3] Giaquinta, M, Multiple integrals in the calculus of variations and nonlinear elliptic systems, (1983), Princeton Univ. Press Princeton · Zbl 0516.49003 [4] Giaquinta, M; Modica, G, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta math., 57, 55-99, (1986) · Zbl 0607.49003 [5] Gilbarg, D; Trudinger, N.S, Elliptic differential equations of second order, (1984), Springer Berlin · Zbl 0691.35001 [6] {\scC. Hamburger}, On the regularity of closed forms minimizing variational integrals, Ph.D. Thesis, Bonn University. [7] Manfredi, J.S, Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations, (1986), Purdue University West Lafayette, preprint [8] Tolksdorff, P, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. mat. pura appl., 134, 241-266, (1983) [9] Tolksdorff, P, Regularity for a more general class of nonlinear elliptic equations, J. differential equations, 51, 126-150, (1984) [10] Uhlenbeck, K, Regularity for a class of nonlinear elliptic systems, Acta math., 138, 219-240, (1977) · Zbl 0372.35030
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