×

Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case. (English) Zbl 1151.35023

Summary: We consider weak solutions of second order nonlinear elliptic systems of divergence type under subquadratic growth conditions. Via the method of \({\mathcal{A}}\)-harmonic approximation we give a characterization of regular points up to the boundary which extends known results from the quadratic and superquadratic case. The proof yields directly the optimal higher regularity on the regular set.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
49N60 Regularity of solutions in optimal control
35D10 Regularity of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Arkhipova A. (2003). Partial regularity up to the boundary of weak solutions of elliptic systems with nonlinearity q greater than two. J. Math. Sci. (N. Y.) 115: 2735–2746 · Zbl 1118.35319
[2] Beck, L.: Partielle Regularität für schwache Lösungen nichtlinearer elliptischer Systeme: der subquadratische Fall. Diploma thesis, Universität Erlangen-Nürnberg (2005)
[3] Campanato S. (1987). Elliptic systems with non-linearity q greater or equal to two. Regularity of the solution of the Dirichlet problem. Ann. Math. Pura Appl. Ser. 4(147): 117–150 · Zbl 0635.35038
[4] Capone C., Greco L. and Iwaniec T. (2002). Higher integrability via Riesz transform and interpolation. Nonlinear Anal. Theory Methods Appl. 49(4(A)): 513–523 · Zbl 1219.35095
[5] Carozza M., Fusco N. and Mingione G. (1998). Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Math. Pura Appl. Ser. 4(175): 141–164 · Zbl 0960.49025
[6] Carozza M. and Mingione G. (2001). Partial regularity of minimizers of quasiconvex integrals with subquadratic growth: the general case. Ann. Pol. Math. 77(3): 219–243 · Zbl 0988.49020
[7] Colombini F. (1971). Un teorema di regolarità alla frontiera per soluzioni di sistemi ellittici quasi lineari. Ann. Sc. Norm. Sup. Pisa Ser. III 25: 15–161 · Zbl 0211.13502
[8] Dacorogna B. (1989). Direct Methods in the Calculus of Variations. Springer-Verlag, Heidelberg · Zbl 0703.49001
[9] de\(\sim\)Giorgi E. (1968). Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. U.M.I. Ser. IV 1: 135–137 · Zbl 0155.17603
[10] Duzaar F. and Grotowski J.F. (2000). Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation. Manus. Math. 103: 267–298 · Zbl 0971.35025
[11] Duzaar F., Grotowski J.F. and Kronz M. (2005). Regularity of almost minimizers of quasi- convex variational integrals with subquadratic growth. Ann. Math. Pura Appl. 11(4): 421–448 · Zbl 1223.49040
[12] Duzaar F., Kristensen J. and Mingione G. (2007). The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math. 602: 17–58 · Zbl 1214.35021
[13] Duzaar F. and Mingione G. (2004). The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. 20: 235–256 · Zbl 1142.35433
[14] Fuchs, M.: Die Green-Matrix für elliptische Systeme zweiter Ordnung in Divergenzform mit stetigen Koeffizienten. Dissertation, Düsseldorf (1983) · Zbl 0554.35041
[15] Giaquinta M. (1983). Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton · Zbl 0516.49003
[16] Giaquinta M. and Modica G. (1979). Almost-everywhere regularity results for solutions of non linear elliptic systems. Manus. Math. 28: 109–158 · Zbl 0411.35018
[17] Giusti E. (2003). Direct Methods in the Calculus of Variation. World Scientific Publishing, Singapore · Zbl 1028.49001
[18] Giusti E. and Misauda M. (1968). Sulla Regolarità delle Soluzioni Deboli di una Classe di Sistemi Ellitici Quasi-lineari. Arch. Rational Mech. Anal. 31: 173–184 · Zbl 0167.10703
[19] Grotowski J.F. (2000). Boundary regularity results for nonlinear elliptic systems in divergence form. Habilitationsschrift, Erlangen
[20] Grotowski J.F. (2002). Boundary regularity results for nonlinear elliptic systems. Calc. Var. 15: 353–388 · Zbl 1148.35315
[21] Hamburger C. (1995). Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. Math. Pura Appl. 169: 321–354 · Zbl 0852.35031
[22] Hamburger, C.: Partial boundary regularity of solutions of nonlinear superelliptic systems. Preprint (2004) · Zbl 1178.35178
[23] Ivert P.-A. (1979). Regularitätsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung. Manus. Math. 30: 53–88 · Zbl 0429.35033
[24] Mingione G. (2006). Regularity of minima: an invitation to the Dark Side of the Calculus of Variations. Appl. Math. 51(4): 355–425 · Zbl 1164.49324
[25] Pepe L. (1971). Risultati di regolarità parziale per le soluzioni \({\mathcal{H}}^{1,p}\) 1 < p < 2 di sistemi ellittici quasi lineari Ann. Univ. Ferrara N. Ser. Sez. VII 16: 129–148
[26] Simon L. (1996). Theorems on Regularity and Singularity of Energy Minimizing Maps. Birkhäuser-Verlag, Basel · Zbl 0864.58015
[27] Wolf, J.: Regularität schwacher Lösungen nichtlinearer elliptischer und parabolischer Systeme partieller Differentialgleichungen mit Entartung. Der Fall 1 < p < 2. Dissertation, Berlin (2001)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.