Regularity theory of the gradient for general nonlinear parabolic systems

References

• Marino , M and Maugeri , A . 2008 . Generalized Gagliardo-Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems . J. Global Optim. , 40 ( 1–3 ) : 185 – 196 .
   Web of Science ®Google Scholar
• Marino , M and Maugeri , A . 1986 . Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth . Rend. Sem. Mat. Padova , 76 : 219 – 245 .
   Google Scholar
• Marino , M and Maugeri , A . 1996 . A remark on the Note “Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth” . Rend. Sem. Mat. Univ. Padova , 95 : 23 – 28 .
   Google Scholar
• Duzaar , F , Mingione , G and Steffen , K . 2011 . Parabolic systems with polynomial growth and regularity , 214 (1005) : Memoirs of the American Mathematical Society . 118 pp
   Google Scholar
• Duzaar , F and Gastel , A . 2002 . Nonlinear elliptic systems with Dini continuous coefficients . Arch. Math. (Basel) , 78 : 58 – 73 .
   Web of Science ®Google Scholar
• Giuffrè , S and Idone , G . 2008 . Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity q > 1 and with natural growth . J. Global Optim. , 40 ( 1–3 ) : 99 – 117 .
   Web of Science ®Google Scholar
• Giuffrè , S . 2004 . Global Hölder regularity for discontinuous elliptic equations in the plane . Proc. Am. Math. Soc. , 132 ( 5 ) : 1333 – 1344 .
   Web of Science ®Google Scholar
• Giuffrè , S and Idone , G . 2011 . On linear and nonlinear boundary value problems in the plane with discontinuous coefficients . Discrete Contin. Dyn. Syst. , 31 ( 4 ) : 1347 – 1363 .
   Web of Science ®Google Scholar
• Ambrosio , L , Caffarelli , L and Maugeri , A . A beautiful walk in the way of the understanding, Discrete Contin. Dyn. Syst. 31(4) (2011), pp. I–VI
   Google Scholar
• Arkhipova , AA . 2000 . Cauchy–Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities. I. On the continuability of smooth solutions . Comment. Math. Univ. Carolin. , 41 ( 4 ) : 693 – 718 .
   Google Scholar
• Arkhipova , AA . On the Global in Time Solvability of the Cauchy-Dirichlet Problem to Nondiagonal Parabolic Systems with Quadratic Nonlinearities, Partial Differential Equations (Praha, 1998), 1–9, Research Notes in Mathematics, Vol. 406, Chapman and Hall/CRC, Boca Raton, FL, 2000
   Google Scholar
• Arkhipova , AA . 1999 . Time-local and time global solvability of the Cauchy-Dirichlet problem for a class of nonlinear non-diagonal parabolic systems . (Russian) Algebra i Analiz , 11 ( 6 ) : 69 – 102 . translation in St. Petersburg Math. J. 11(6) (2000), pp. 989–1017
   Google Scholar
• Arkhipova , AA . 2001 . Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic nonlinearities. II. Local and global solvability results . Comment. Math. Univ. Carolin. , 42 ( 1 ) : 53 – 76 .
   Google Scholar
• Marino , M and Maugeri , A . 1989 . Partial Hölder continuity of solutions of nonlinear parabolic systems of second order with quadratic growth . Boll. Un. Mat. Ital. B(7) , 3 ( 2 ) : 397 – 435 .
   Google Scholar
• Fattorusso , L . 1990 . Un risultato di differenziabilità per sistemi parabolici non lineari in ipotesi di monotonia . Rend. Circolo Mat. Palermo , 39 : 412 – 426 .
   Google Scholar
• Campanato , S . Sistemi ellittici in forma divergenza. Regolarità all'interno, Quaderni Scuola Normale Sup. Pisa, 1980, 187 pp
   Google Scholar
• Campanato , S . 1981 . Partial Hölder continuity of solutions of quasilinear parabolic systems of second order with linear growth . Rend. Sem. Mat. Padova , 64 : 59 – 75 .
   Google Scholar
• Campanato , S . 1984 . On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions . Ann. Mat. Pura Appl. (4) , 137 : 83 – 122 .
   Web of Science ®Google Scholar
• Giusti , E . 1969 . Precisazione delle funzioni H 1,p e singolarità delle soluzioni deboli di sistemi ellittici non lineari . Boll. Un. Mat. Ital. , 2 : 71 – 76 .
   Google Scholar