Abstract
In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.
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Rockafellar, R.T.: The theory of subgradients and its applications to problems of optimization, volume 1 of R & E. Heldermann, Berlin. Convex and nonconvex functions (1981)
Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969)
Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50) (1973)
Chill, R., Mildner, S.: The Kurdyka-łojasiewicz-Simon inequality and stabilisation in nonsmooth infinite-dimensional gradient systems. Proc. Amer. Math. Soc. 146(10), 4307–4314 (2018)
Cipriani, F., Grillo, G.: Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces. J. Reine Angewandte Math. (Crelle’s J.) 562, 201–235 (2003)
Barthélemy, L.: Invariance d’un convexe fermé par un semi-groupe associé à une forme non-linéaire. Abstr. Appl. Anal. 1(3), 237–262 (1996)
Bénilan, P., Crandall, M.G.: Completely accretive operators. In: Semigroup theory and evolution equations (Delft, 1989), Lecture Notes in Pure and Appl. Math., vol. 135, pp. 41–75. Dekker, New York (1991)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter & Co., Berlin. extended edition (2011)
Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space, De Gruyter Studies in Mathematics, vol. 14. Walter de Gruyter & Co., Berlin (1991)
Ma, Z.M., Röckner, M.: Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer, Berlin (1992)
Biroli, M.: Strongly local nonlinear Dirichlet functionals. Ukr. Mat. Visn. 1(4), 485–500 (2004)
Biroli, M., Vernole, P.G.: Strongly local nonlinear Dirichlet functionals and forms. Adv. Math. Sci. Appl. 15(2), 655–682 (2005)
Musielak, J.: Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)
Kaijser, S.: A note on dual Banach spaces. Math. Scand. 41(2), 325–330 (1977)
Bénilan, P., Picard, C.: Quelques aspects non linéaires du principe du maximum. In: Séminaire de Théorie du Potentiel, No. 4 (Paris, 1977/1978), Lecture Notes in Math., vol. 713, pp. 1–37. Springer, Berlin (1979)
Barbu, V., Precupanu, T.: Convexity and optimization in Banach spaces, 4th edn. Springer Monographs in Mathematics. Springer, Dordrecht (2012)
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author would like to thank the referee for the careful reading of the manuscript and the valuable suggestions, which have been very helpful in improving the paper.
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Claus, B. Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting. Potential Anal 58, 159–179 (2023). https://doi.org/10.1007/s11118-021-09935-y
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DOI: https://doi.org/10.1007/s11118-021-09935-y