Boundary estimates for superharmonic functions and solutions of semilinear elliptic equations with source

• Autores: Kentaro Hirata
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 1, 2021, págs. 43-61
• Idioma: inglés
• DOI: 10.1007/s13348-020-00279-1
• Texto completo no disponible (Saber más ...)
• Resumen

  • In a certain Lipschitz domain \Omega \subset {\mathbb {R}}^n, we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality -\Delta u\le cu^p, where c>0 and 1 p (n+\alpha )/(n+\alpha -2) with constant \alpha regarding the lower bound estimate of the Green function on \Omega. An argument combined estimates for certain Green potentials and iteration methods enables us to prove it. Results are applicable to positive solutions of semilinear elliptic equations like -\Delta u=a(x)u^p with a(x) being nonnegative and bounded on \Omega. Also, we present an a priori estimate and a removability theorem for positive solutions having isolated singularities at a boundary point. The former extends one given by Bidaut-Véron and Vivier (Rev Mat Iberoam 16:477–513, 2000) in the case where \Omega has a smooth boundary and a(x)\equiv 1.

• Referencias bibliográficas
  • Aikawa, H.: Boundary Harnack principle and Martin boundary for a uniform domain. J. Math. Soc. Japan 53, 119–145 (2001)
  • Aikawa, H.: Potential-theoretic characterizations of nonsmooth domains. Bull. Lond. Math. Soc. 36, 469–482 (2004)
  • Aikawa, H., Hirata, K., Lundh, T.: Martin boundary points of a John domain and unions of convex sets. J. Math. Soc. Japan 58, 247–274 (2006)
  • Aikawa, H., Lundh, T., Mizutani, T.: Martin boundary of a fractal domain. Potential Anal. 18, 311–357 (2003)
  • Aikawa, H., Kilpeläinen, T., Shanmugalingam, N., Zhong, X.: Boundary Harnack principle for p-harmonic functions in smooth Euclidean domains....
  • Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst....
  • Ancona, A.: Une propriété de la compactification de Martin d’un domaine euclidien. Ann. Inst. Fourier (Grenoble) 29, 71–90 (1979)
  • Bauman, P.: Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22, 153–173 (1984)
  • Bidaut-Véron, M.F., Borghol, R., Véron, L.: Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic...
  • Bidaut-Véron, M.F., Vivier, L.: An elliptic semilinear equation with source term involving boundary measures: the subcritical case. Rev. Mat....
  • Bogdan, K.: Sharp estimates for the Green function in Lipschitz domains. J. Math. Anal. Appl. 243(2), 326–337 (2000)
  • Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana...
  • Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65, 275–288 (1977)
  • Hirata, K.: Global estimates for non-symmetric Green type functions with applications to the p-Laplace equation. Potential Anal. 29, 221–239...
  • Hirata, K.: Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains. J. Math. Soc. Japan 62, 1043–1068...
  • Hirata, K.: Positive solutions with a time-independent boundary singularity of semilinear heat equations in bounded Lipschitz domains. Nonlinear...
  • Hirata, K.: Two-sided estimates for positive solutions of superlinear elliptic boundary value problems. Bull. Aust. Math. Soc. 98, 465–473...
  • Jerison, D.S., Kenig, C.E.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46, 80–147 (1982)
  • Kemper, J.T.: A boundary Harnack principle for Lipschitz domains and the principle of positive singularities. Commun. Pure Appl. Math. 25,...
  • Lewis, J.L., Nyström, K.: Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains. Ann. Sci. École Norm....
  • Maeda, F.-Y., Suzuki, N.: The integrability of superharmonic functions on Lipschitz domains. Bull. Lond. Math. Soc. 21, 270–278 (1989)
  • Marcus, M., Véron, L.: Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: the subcritical case. Ann....
  • McKenna, P.J., Reichel, W.: A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains. J. Funct....
  • Naïm, L.: Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel. Ann. Inst. Fourier Grenoble 7, 183–281 (1957)
  • Poláčik, P., Quittner, P., Souplet, P.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations...
  • Wu, J.M.G.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier...