Regularity for minima of functionals with p-growth. (English) Zbl 0674.35008
We prove that the first derivatives of scalar minima of functionals of the type
\[
I(u)=\int f(x,u,\nabla u)dx,
\]
are Hölder continuous. Here \(f(x,u,\nabla u)\approx | \nabla u|^ p,\) \(1<p<\infty\) and f is assumed Hölder continuous in x and u.
We give two applications. One to the regularity theory of quasiregular mappings and the other to quasilinear degenerate elliptic equations with p-growth.
We give two applications. One to the regularity theory of quasiregular mappings and the other to quasilinear degenerate elliptic equations with p-growth.
Reviewer: J.J.Manfredi
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35J70 | Degenerate elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35A15 | Variational methods applied to PDEs |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
Keywords:
first derivatives; scalar minima; Hölder continuous; regularity; quasiregular mappings; quasilinear; p-growthReferences:
| [1] | Bojarski, B.; Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in \(ofR^n\), Ann. Acad. Sci. Fenn. Ser. AI Math., 8, 257-324 (1983) · Zbl 0548.30016 |
| [2] | Campanato, S., Equazione ellittiche del secondo ordine e spazi \(L^{2,λ}\), Ann. Mat. Pura Appl., 69, 321-380 (1965) · Zbl 0145.36603 |
| [3] | DiBenedetto, E., \(C^{1 + α}\) Local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.: Theory, Methods Appl., 7, 827-850 (1983) · Zbl 0539.35027 |
| [4] | Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, (Annals of Math. Studies, Vol. 105 (1983), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ) · Zbl 1006.49030 |
| [5] | Giaquinta, M.; Giusti, E., Differentiability of minima of non-differentiable functionals, Invent. Math., 72, 285-298 (1983) · Zbl 0513.49003 |
| [6] | Giaquinta, M.; Giusti, E., On the regularity of the minima of variational integrals, Acta Math., 148, 31-46 (1982) · Zbl 0494.49031 |
| [7] | Granlund, S.; Lindqvist, P.; Martio, O., Conformally invariant variational integrals, Trans. Amer. Math. Soc., 277, 43-73 (1983) · Zbl 0518.30024 |
| [8] | Giaquinta, M.; Modica, G., Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57, 55-99 (1986), Preprint · Zbl 0607.49003 |
| [9] | Gilbart, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin/Heidelberg/New York · Zbl 0361.35003 |
| [10] | Iwaniec, T., Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions, Dissertationes Math., 198 (1982) · Zbl 0524.35019 |
| [11] | Lewis, J., Regularity of the derivatives of solutions to certain elliptic equations, Indiana Univ. Math. J., 32, 849-858 (1983) · Zbl 0554.35048 |
| [12] | Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations (1968), Academic Press: Academic Press New York · Zbl 0164.13002 |
| [13] | Manfredi, J., Regularity of the Gradient for a Class of Nonlinear Possibly Degenerate Elliptic Equations, (Ph.D. thesis (1986), Washington University: Washington University Saint Louis) |
| [14] | Tolksdorff, P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51, 126-150 (1984) · Zbl 0488.35017 |
| [15] | Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems, Acta Math., 138, 219-240 (1977) · Zbl 0372.35030 |
| [16] | Ural’tseva, N., Degenerate Quasilinear Elliptic Systems, (Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov, 7 (1968)), 184-222, [In Russian] · Zbl 0199.42502 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
