Nonlocal Equations with Measure Data
Tóm tắt
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149–169, 1989, Partial Differ Equ 17:641–655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591–613, 1992, Acta Math 172:137–161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón–Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321–381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.
Tài liệu tham khảo
Alberti G., Bellettini G.: A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann. 310, 527–560 (1998)
Alibaud N., Andreianov B., Bendahmane M.: Renormalized solutions of the fractional Laplace equation. C. R. Math. Acad. Sci. Paris 348, 759–762 (2010)
Barles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13, 1–26 (2011)
Bénilan P., Boccardo L., Gallouët T., Gariepy R., Pierre M., Vázquez J.L.: An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 22, 241–273 (1995)
Bjorland C., Caffarelli L., Figalli A.: Non-local tug-of-war and the infinity fractional Laplacian. Adv. Math. 230, 1859–1894 (2012)
Boccardo L., Gallouët T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)
Boccardo L., Gallouët T.: Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17, 641–655 (1992)
Boccardo L., Gallouët T.: W 1,1-solutions in some borderline cases of Calderón–Zygmund theory. J. Differ. Equ. 253, 2698–2714 (2012)
Boccardo L., Gallouët T., Orsina L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincarè Anal. Non Linéaire 13, 539–551 (1996)
Bonforte M., Väzquez J.L.: Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014)
Cabré X., Sola-Morales X.: Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58, 1678–1732 (2005)
Cacace S., Garroni A.: A multi-phase transition model for the dislocations with interfacial microstructure. Interfaces Free Bound 11, 291–316 (2009)
Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. PDE 32, 1245–1260 (2007)
Caffarelli L., Silvestre L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011)
Caffarelli L., Vasseur A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (II) 171, 1903–1930 (2010)
Campanato S.: Equazioni ellittiche del II ordine e spazi \({\mathfrak{L}^{(2,\lambda)}}\). Ann. Mat. Pura Appl. (IV) 69, 321–381 (1965)
Chen H., Veron L.: Semilinear fractional elliptic equations involving measures. J. Differ. Equ. 25, 565–590 (1973)
Cianchi A.: Non-linear potentials, local solutions to elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. (V) 10, 335–361 (2011)
Dal Maso G., Murat F., Orsina L., Prignet A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 28, 741–808 (1999)
Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional p-minimizers. (submitted paper)
Di Castro A., Kuusi T., Palatucci G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267, 1807–1836 (2014)
Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Duzaar F., Mingione G.: Gradient estimates via non-linear potentials. Am. J. Math. 133, 1093–1149 (2011)
Gilboa G., Osher S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)
Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc, River Edge (2003)
Havin M., Maz’ja V.G.: Non-linear potential theory. Russ. Math. Surv. 27, 71–148 (1972)
Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, New York (1993)
Jaye B., Verbitsky I.: Local and global behaviour of solutions to nonlinear equations with natural growth terms. Arch. Ration. Mech. Anal. 204, 627–681 (2012)
Karlsen K.H., Petitta F., Ulusoy S.: A duality approach to the fractional Laplacian with measure data. Publ. Mat. 55, 151–161 (2011)
Kilpeläinen T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures. Potential Anal. 3, 265–272 (1994)
Kilpeläinen T., Kuusi T., Tuhola-Kujanpää A.: Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincaré, Anal. Non Linèaire 28, 775–795 (2011)
Kilpeläinen T., Malý J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 19, 591–613 (1992)
Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)
Klimsiak T., Rozkosz A.: Dirichlet forms and semilinear elliptic equations with measure data. J. Funct. Anal. 265, 890–925 (2013)
Korte R., Kuusi T.: A note on the Wolff potential estimate for solutions to elliptic equations involving measures. Adv. Calc. Var. 3, 99–113 (2010)
Kuusi T., Mingione G.: Universal potential estimates. J. Funct. Anal. 262, 4205–4269 (2012)
Kuusi T., Mingione G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4, 1–82 (2014)
Lieberman G.M.: Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures. Commun. Partial Differ. Equ. 18, 1191–1212 (1993)
Lindqvist P.: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. (Crelles J.) 365, 67–79 (1986)
Maz’ya V.: The continuity at a boundary point of the solutions of quasi-linear elliptic equations. (Russian) Vestn. Leningr. Univ. 25, 42–55 (1970)
Mingione G.: Nonlinear measure data problems. Milan J. Math. 79, 429–496 (2011)
Phuc N.C., Verbitsky I.E.: Quasilinear and Hessian equations of Lane–Emden type. Ann. Math. (II) 168, 859–914 (2008)
Phuc N.C., Verbitsky I.E.: Singular quasilinear and Hessian equations and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)
Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematics Series, vol. 32. Princeton Univ. Press, Princeton (1971)
Trudinger N.S., Wang X.J.: On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124, 369–410 (2002)
Vázquez J.L.: Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J. Eur. Math. Soc. (JEMS) 16, 769–803 (2014)
Vázquez, J.L.: Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. DCDS (to appear)