Global Hölder regularity for discontinuous elliptic equations in the plane

Giuffrè, Sofia

doi:10.1090/S0002-9939-03-07348-9

Global Hölder regularity for discontinuous elliptic equations in the plane
HTML articles powered by AMS MathViewer

by Sofia Giuffrè PDF

Proc. Amer. Math. Soc. 132 (2004), 1333-1344 Request permission

Abstract:

$C^{1, \mu }$-regularity up to the boundary is proved for solutions of boundary value problems for elliptic equations with discontinuous coefficients in the plane. In particular, we deal with the Dirichlet boundary condition \begin{equation*} \begin {array}{ll} u= g(x) &\ \rm on \: \partial \Omega \end{array} \end{equation*} where $g(x) \in W^{2- \frac {1}{r}, r}(\partial \Omega )$, $r>2$, or with the following normal derivative boundary conditions: \begin{equation*} \begin {array}{lclr} \displaystyle \frac {\partial u}{\partial n} = h( x) &\ \rm or &\ \displaystyle \frac {\partial u}{\partial n} + \sigma u = h( x) &\ \rm on \: \partial \Omega \end{array} \end{equation*} where $h(x) \in W^{1- \frac {1}{r}, r}(\partial \Omega )$, $r>2$, $\sigma >0$ and $n$ is the unit outward normal to the boundary $\partial \Omega$.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
S. Campanato, Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 701–707 (Italian). MR 224996
Robert Finn and James Serrin, On the Hölder continuity of quasi-conformal and elliptic mappings, Trans. Amer. Math. Soc. 89 (1958), 1–15. MR 97626, DOI 10.1090/S0002-9947-1958-0097626-4
Sofia Giuffrè, Oblique derivative problem for nonlinear elliptic discontinuous operators in the plane with quadratic growth, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 10, 859–864 (English, with English and French summaries). MR 1689841, DOI 10.1016/S0764-4442(99)80286-9
Enrico Giusti, Sulla regolarità delle soluzioni di una classe di equazioni ellittiche, Rend. Sem. Mat. Univ. Padova 39 (1967), 362–375 (Italian). MR 226173
Philip Hartman, Hölder continuity and non-linear elliptic partial differential equations, Duke Math. J. 25 (1958), 57–65. MR 97625
A. Maugeri, D. K. Palagachev and L. Softova, Elliptic and parabolic equations with discontinuous coefficients, Wiley, VCH Publishers, 2000.
Carlo Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl. (4) 63 (1963), 353–386 (Italian). MR 170090, DOI 10.1007/BF02412185
C. B. Morrey, On the solutions of quasilinear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
Dian K. Palagachev, Global strong solvability of Dirichlet problem for a class of nonlinear elliptic equations in the plane, Matematiche (Catania) 48 (1993), no. 2, 311–321 (1994). MR 1320671
L. Softova, An integral estimate for the gradient for a class of nonlinear elliptic equations in the plane, Z. Anal. Anwendungen 17 (1998), no. 1, 57–66. MR 1616060, DOI 10.4171/ZAA/808
Giorgio Talenti, Equazioni lineari ellittiche in due variabili, Matematiche (Catania) 21 (1966), 339–376 (Italian). MR 204845
Giorgio Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl. (4) 69 (1965), 285–304 (Italian). MR 201816, DOI 10.1007/BF02414375
N. S.Trudinger, Nonlinear second order elliptic equations, Lecture Notes of Math. Inst. of Nankai Univ., Tianjin, China, 1986.