Abstract
In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to \(L^2(-a, 0, H^2(B(\sigma), {\mathbb{R}}^N)) \cap H^1(-a, 0, L^2(B(\sigma), {\mathbb{R}}^N))\) (see Theorem 3).
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References
Campanato S. and Cannarsa P. (1981). Differentiability and partial Hölder continuity of the solutions of non-linear elliptic systems of order 2m with quadratic growth. Ann. Scuola Norm. Pisa 8: 285–309
Fattorusso L. (1987). Sulla differenziabilità delle soluzioni di sistemi parabolici non lineari del secondo ordine ad andamento quadratico. Boll. U.M.I. 1-B: 741–764
Fattorusso L. (1990). Un risultato di differenziabilità per sistemi parabolici non lineari in ipotesi di monotonia. Rend. Circ. Matem. Palermo 39: 412–426
Marino M. and Maugeri A. (1989). Partial Hölder continuity of solutions of nonlinear parabolic systems of second order with quadratic growth. Boll. U.M.I. 3-B: 397–435
Marino M. and Maugeri A. (1991). Régularité et principe du maximum pour la solution du problème de Cauchy-Dirichlet inhomogène pour des systèmes paraboliques. C.R. Acad. Sci. Paris 312: 277–279
Marino M. and Maugeri A. (1991). Maximum principle for parabolic systems. Algebra i Analiz 3: 155–163
Marino M. and Maugeri A. (1993). L 2, λ regularity of the spatial derivatives of the solutions to parabolic systems in divergence form. Ann. Mat. Pura Appl. 164: 275–298
Marino M. and Maugeri A. (1995). Differentiability of weak solutions of nonlinear parabolic systems with quadratic growth. Mathematiche (Catania) 50: 361–377
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
Miranda C. (1965). Su alcuni teoremi di inclusione. Ann. Polonici Math. 16: 305–315
Naumann J. (1990). On the interior differentiability of weak solutions of parabolic systems with quadratic growth nonlinearities. Rend. Sem. Mat. Univ. Padova 83: 55–70
Naumann J. and Wolf J. (1997). Interior differentiability of weak solutions of parabolic systems with quadratic growth nonlinearities. Rend. Sem. Mat. Univ. Padova 98: 253–272
Nirenberg L. (1966). An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa 20: 733–737
Triebel H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publ. Comp., Amsterdam, New York, Oxford
Triebel H. (1983). Theory of Function Spaces. Birkhäuser Verlag, Basel, Boston, Stuttgart
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Marino, M., Maugeri, A. Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems. J Glob Optim 40, 185–196 (2008). https://doi.org/10.1007/s10898-007-9184-7
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DOI: https://doi.org/10.1007/s10898-007-9184-7