Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in \(\mathbb {R}^N\) and applications

1. Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In: Methods of Functional Analysis and Theory of Elliptic Equations, Naples: 19–52, p. 1983. Liguori, Naples (1982)

2. Aronsson, G.: Minimization problems for the functional \(\sup _x F (x, f (x), f^{\prime }(x))\). Ark. Mat. 6, 33–53 (1965)

   Article  MathSciNet  Google Scholar 

3. Aronsson, G.: Minimization problems for the functional \(\sup _x F (x, f (x), f^{\prime }(x))\). II. Ark. Mat. 6, 409–431 (1966)

   Article  Google Scholar 

4. Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)

   Article  MathSciNet  Google Scholar 

5. Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. (N.S.) 41(4), 439–505 (2004)

6. Araújo, D.J., Leitão, R., Teixeira, E.V.: Infinity Laplacian equation with strong absorptions. J. Funct. Anal. 270, 2249–2267 (2016)

   Article  MathSciNet  Google Scholar 

7. Biswas, A., Roychowdhury, P.: Generalized principal eigenvalues of convex nonlinear elliptic operators in \({\mathbb{R}}^N\). Adv. Calculus Var. (to appear) (2020)

8. Biswas, A., Vo, H.-H.: Liouville theorems for infinity Laplacian with gradient and KPP type equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.202105_050

9. Armstrong, S.N., Smart, C.K., Somersille, S.J.: An infinity Laplace equation with gradient term and mixed boundary conditions. Proc. Am. Math. Soc. 139(5), 1763–1776 (2011)

   Article  MathSciNet  Google Scholar 

10. Berestycki, H., Nirenberg, L., Varadhan, S.R.S.: The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Commun. Pure Appl. Math. 47(1), 47–92 (1994)

    Article  MathSciNet  Google Scholar 

11. Berestycki, H., Capuzzo Dolcetta, I., Porretta, A., Rossi, L.: Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators. J. Math. Pures Appl. 103, 1276–1293 (2015)

    Article  MathSciNet  Google Scholar 

12. Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. II. General domains. J. Am. Math. Soc. 23, 1–34 (2010)

13. Berestycki, H., Hamel, F., Rossi, L.: Liouville-type results for semilinear elliptic equations in unbounded domains. Ann. Mat. Pura Appl. (4) 186(3), 469–507 (2007)

14. Berestycki, H., Rossi, L.: Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains. Commun. Pure Appl. Math. 68(6), 1014–1065 (2015)

    Article  MathSciNet  Google Scholar 

15. Berestycki, H., Rossi, L.: On the principal eigenvalue of elliptic operators in \(\mathbb{R}^N\) and applications. JEMS 8, 195–215 (2006)

    Article  MathSciNet  Google Scholar 

16. Bhattacharya, T., Marazzi, L.: An eigenvalue problem for the infinity-Laplacian. Electron. J. Differ. Equ. 47,(2013)

17. Bhattacharya, T.: An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions. Electron. J. Differ. Equ. 44,(2001)

18. Bhattacharya, T.: A Boundary Harnack Principle for Infinity-Laplacian and Some Related Results. Boundary Value Problems (2007)

19. Birindelli, I., Demengel, F.: First eigenvalue and maximum principle for fully nonlinear singular operators. Adv. Differ. Equ. 11(1), 91–119 (2006)

    MathSciNet  MATH  Google Scholar 

20. Caselles, V., Morel, J.-M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7, 376–386 (1998)

    Article  MathSciNet  Google Scholar 

21. Charro, F., Peral, I.: Zero order perturbations to fully nonlinear equations: comparison, existence and uniqueness. Commun. Contemp. Math. 11, 131–164 (2009)

    Article  MathSciNet  Google Scholar 

22. Crandall, M. G.: A visit with the \(\infty \)-Laplace equation. In: Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Math., vol. 1927, pp. 75–122. Springer, Berlin (2008)

23. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second-order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

    Article  MathSciNet  Google Scholar 

24. da Silva, J.V., Rossi, J., Salort, A.M.: Maximal solutions for the \(\infty \)-eigenvalue problem. Adv. Calc. Var. 12(2), 181–191 (2019)

    Article  MathSciNet  Google Scholar 

25. D’Onofrio, L., Giannetti, F., Iwaniec, T., Manfredi, J., Radice, T.: Divergence forms of the \(\infty \)-Laplacian. Publ. Mat. 50(1), 229–248 (2006)

    Article  MathSciNet  Google Scholar 

26. Evans, L.: Estimates for smooth absolutely minimizing Lipschitz extensions. Electronic J. Differ. Equ. 3, 1–10 (1993)

    MathSciNet  MATH  Google Scholar 

27. Furusho, Y., Ogura, Y.: On the existence of bounded positive solutions of semilinear elliptic equations in exterior domains. Duke Math. J. 48(3), 497–521 (1981)

    Article  MathSciNet  Google Scholar 

28. Juutinen, P.: Principal eigenvalue of a very badly degenerate operator and applications. J. Differ. Equ. 236(2), 532–550 (2007)

    Article  MathSciNet  Google Scholar 

29. Juutinen, P., Lindqvist, P.: On the higher eigenvalues for the \(\infty \)-eigenvalue problem. Calc. Var. Partial Differ. Equ. 23(2), 169–192 (2005)

30. Juutinen, P., Lindqvist, P., Manfredi, J.J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2), 89–105 (1999)

    Article  MathSciNet  Google Scholar 

31. Lindqvist, P.: Notes on the infinity Laplace equation. Springer Briefs in Mathematics. BCAM Basque Center for Applied Mathematics, Bilbao. Springer, Cham (2016)

32. Lindqvist, P., Manfredi, J.: The Harnack inequality for \(\infty \)-harmonic functions. Electron. J. Differ. Equ. 04,(1995)

33. López-Soriano, R., Navarro-Climent, J.C., Rossi, J.D.: The infinity Laplacian with a transport. J. Math. Anal. Appl. 398(2), 752–765 (2013)

    Article  MathSciNet  Google Scholar 

34. Nussbaum, R. D., Pinchover, Y.: On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. J. Anal. Math. 59:161–177 (Festschrift on the occasion of the 70th birthday of Shmuel Agmon) (1992)

35. Nguyen, P.-T., Vo, H.-H.: On the generalized principal eigenvalue of quasilinear operator: definitions and qualitative properties. Calc. Var. Partial Differ. Equ. 58(3), 102 (2019)

    Article  MathSciNet  Google Scholar 

36. Nguyen, P.-T., Vo, H.-H.: Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations. J. Funct. Anal. 269, 3120–3146 (2015)

    Article  MathSciNet  Google Scholar 

37. Patrizi, S.: The principal eigenvalue of the \(\infty \)-Laplacian with the Neumann boundary condition. ESAIM Control Optim. Calc. Var. 17(2), 575–601 (2011)

    Article  MathSciNet  Google Scholar 

38. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22, 167–210 (2009)

    Article  MathSciNet  Google Scholar 

39. Portilheiro, M., Vázquez, J.L.: A porous medium equation involving the infinity-Laplacian. Viscosity solutions and asymptotic behavior. Commun. Partial Differ. Equ. 37(5), 753–793 (2012)

40. Portilheiro, M., Vázquez, J.L.: Degenerate homogeneous parabolic equations associated with the infinity-Laplacian. Calc. Var. Partial Differ. Equ. 46(3–4), 705–724 (2013)

    Article  MathSciNet  Google Scholar 

41. Yu, Y.: Some properties of the ground states of the infinity Laplacian. Indiana Univ. Math. J. 56, 947–964 (2007)

    Article  MathSciNet  Google Scholar 

Download references