Abstract
It is known that the De Giorgi’s conjecture does not hold in two dimensions for semilinear elliptic equations with a nonzero drift, in general,
when \(q=(0,-c)\) for \(c\ne 0\). This equation arises in the modeling of Bunsen burner flames. Bunsen flames are usually made of two flames: a diffusion flame and a premixed flame. In this article, we prove De Giorgi type results, and stability conjecture, for the following local-nonlocal counterpart of the above equation (with a nonlocal premixed flame) in two dimensions,
when L is a nonlocal operator, \(f\in C^1({\mathbb {R}})\) and \(c\in {\mathbb {R}}^+\). In addition, we provide a priori estimates for the above equation, when \(n\ge 1\), with various jumping kernels. The operator \(\Delta +cL\) is an infinitesimal generator of jump-diffusion processes in the context of probability theory.
Similar content being viewed by others
References
Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65, 9–33 (2001)
Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in \({\mathbb{R} }^3\) and a conjecture of De Giorgi. J. Amer. Math. Soc. 13, 725–739 (2000)
Barlow, M.T.: On the Liouville property for divergence form operators. Can. J. Math. 50, 487–496 (1998)
Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361, 1963–1999 (2009)
Barlow, M.T., Bass, R.F., Gui, C.: The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math. 53, 1007–1038 (2000)
Bass, R.F., Kumagai, T., Uemura, T.: Convergence of symmetric Markov chains on \({\mathbb{Z} }^d\). Probab. Theory Relat. Fields 148, 107–140 (2010)
Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potent. Anal. 17, 375–388 (2002)
Berestycki, H., Hamel, F., Monneau, R.: One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 103, 375–396 (2000)
Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, 69–94 (1997)
Bonnet, A., Hamel, F.: Existence of nonplanar solutions of a simple model of premixed Bunsen flames. SIAM J. Math. Anal. 31, 80–118 (1999)
Cabré, X., Cinti, E.: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Dis. Contin. Dyn. Syst. 28, 1179–1206 (2010)
Cabré, X., Cinti, E.: Sharp energy estimates for nonlinear fractional diffusion equations. Calc. Var. Partial. Differ. Equ. 49, 233–269 (2014)
Cabré, X., Serra, J.: An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions. Nonlinear Anal. 137, 246–265 (2016)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 23–53 (2014)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Amer. Math. Soc. 367, 911–941 (2015)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Differ. Equ. 32, 1245–1260 (2007)
Cabré, X., Solá-Morales, J.: Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58, 1678–1732 (2005)
Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23, 245–287 (1987)
Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components. J. Reine Angew. Math. 711, 111–138 (2016)
Chen, Z.-Q., Kim, P., Song, R., Vondracek, Z.: Boundary Harnack principle for \(\Delta + \Delta ^{\frac{\alpha }{2}}\). Trans. Amer. Math. Soc. 364, 4169–4205 (2012)
Chen, Z.Q., Kim, P., Song, R.: Heat kernel estimates for \(\Delta + \Delta ^{\frac{\alpha }{2}}\) in \(C^{1,1}\) open sets. J. Lond. Math. Soc. 2(84), 58–80 (2011)
Chen, Z.-Q., Kim, P., Song, R., Vondracek, Z.: Sharp green function estimates for \(\Delta + \Delta ^{\frac{\alpha }{2}}\) in \(C^{1,1}\) open sets and their applications. Illinois J. Math. 54, 981–1024 (2010)
Chen, Z.-Q., Kim, P., Kumagai, T.: Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math. Ann. 342, 833–883 (2008)
Chen, Z.-Q., Kim, P., Kumagai, T.: Global heat kernel estimates for symmetric jump processes. Trans. Amer. Math. Soc. 363, 5021–5055 (2011)
Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for \(\Delta ^{\frac{\alpha }{2}} + \Delta ^{\frac{\beta }{2}}\). Ill. J. Math. 54, 1357–1392 (2010)
Chen, Z.-Q., Kumagai, T.: A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev. Mat. Iberoam. 26, 551–589 (2010)
Chen, Z.-Q., Rohde, S.: Schramm-Loewner equations driven by symmetric stable processes. Comm. Math. Phys. 285, 799–824 (2009)
Cinti, E., Ferrari, F.: Geometric inequalities for fractional Laplace operators and applications. Nonlinear Differ. Equ. Appl. (NoDEA) 22, 1699–1714 (2015)
Cozzi, M., Passalacqua, T.: One-dimensional solutions of non-local Allen–Cahn-type equations with rough kernels. J. Differ. Equ. 260, 6638–6696 (2016)
De Giorgi, E.: Convergence problems for functional and operators. In: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978). Pitagora, Bologna, pp. 131–188 (1979)
del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi’s conjecture in dimension \(N\ge 9\). Ann. Math. 2(174), 1485–1569 (2011)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Farina, A.: Symmetry for solutions of semilinear elliptic equations in \({\mathbb{R} }^n\) and related conjectures. Ricerche Math. 48, 129–154 (1999)
Farina, A., Sciunzi, B., Valdinoci, E.: Bernstein and de giorgi type problems: new results via a geometric approach. Ann. Sc. Norm. Super. Pisa Cl. Sci. 7, 741–791 (2008)
Fazly, M.: Higher-dimensional solutions for a nonuniformly elliptic equation. Int. Math. Res. Not. (IMRN) 5, 1315–1337 (2015)
Fazly, M., Ghoussoub, N.: De Giorgi type results for elliptic systems. Calc. Var. Part. Differ. Equ. 47, 809–823 (2013)
Fazly, M., Gui, C.: On nonlocal systems with jump processes of finite range and with decays. J. Differ. Equ. 268(6), 3171–3200 (2020)
Fazly, M., Sire, Y.: Symmetry properties for solutions of nonlocal equations involving nonlinear operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 36, 523–543 (2019)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, p 392 (1994)
Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)
Ghoussoub, N., Gui, C.: About De Giorgi’s conjecture in dimensions 4 and 5. Ann. Math. 2(157), 313–334 (2003)
Graham, C.: Nonlinear diffusion with jumps. Ann. Inst. H. Poincaré Probab. Statist. 28, 393–402 (1992)
Gui, C.: Symmetry of traveling wave solutions to the Allen–Cahn equation in \({\mathbb{R} }^2\). Arch. Ration. Mech. Anal. 203, 1037–1065 (2012)
Hamel, F., Ros-Oton, X., Sire, Y., Valdinoci, E.: A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 469–482 (2017)
Modica, L.: A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math. 38, 679–684 (1985)
Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. 192, 673–718 (2013)
Savin, O.: Regularity of flat level sets in phase transitions. Ann. Math. 2(169), 41–78 (2009)
Silvestre, L.: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55, 1155–1174 (2006)
Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256, 1842–1864 (2009)
Song, R., Vondracek, Z.: Harnack inequality for some classes of Markov processes. Math. Z. 246, 177–202 (2004)
Sternberg, P., Zumbrun, K.: A Poincaré inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. Math. 503, 63–85 (1998)
Sternberg, P., Zumbrun, K.: Connectivity of phase boundaries in strictly convex domains. Arch. Ration. Mech. Anal. 141, 375–400 (1998)
Acknowledgements
The author would like to thank Professor Yannick Sire for discussions and comments on this topic. The author is thankful to anonymous referee(s) for valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Fazly, M. De Giorgi type results for equations with nonlocal lower-order terms. Annali di Matematica 202, 519–550 (2023). https://doi.org/10.1007/s10231-022-01251-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-022-01251-5
Keywords
- De Giorgi’s conjecture
- Jump-diffusion processes
- Local and nonlocal operators
- Stable solutions
- A priori estimates