Figure 1.
The unit outward normal $ \mathbf{n} $ and the conormal $ \boldsymbol\nu $ to $ \partial \Omega $
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May
2020, 13(5): 1601-1618.
doi: 10.3934/dcdss.2020091
The hypoelliptic Robin problem for quasilinear elliptic equations
Institute of Mathematics, University of Tsukuba, Tsukuba 305–8571, Japan |
This paper is devoted to the study of a hypoelliptic Robin boundary value problem for quasilinear, second-order elliptic differential equations depending nonlinearly on the gradient. More precisely, we prove an existence and uniqueness theorem for the quasilinear hypoelliptic Robin problem in the framework of Hölder spaces under the quadratic gradient growth condition on the nonlinear term. The proof is based on the comparison principle for quasilinear problems and the Leray–Schauder fixed point theorem. Our result extends earlier theorems due to Nagumo, Akô and Schmitt to the hypoelliptic Robin case which includes as particular cases the Dirichlet, Neumann and regular Robin problems.
Keywords:
Quasilinear elliptic equation,
hypoelliptic Robin problem,
Nagumo condition,
comparison principle,
Leray–Schauder fixed point theorem.
Mathematics Subject Classification:
Primary: 35J62; Secondary: 35H10, 35R25.
Citation:
Kazuaki Taira. The hypoelliptic Robin problem for quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S,
2020, 13
(5)
: 1601-1618.
doi: 10.3934/dcdss.2020091
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References:
| [1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Second edition, Elsevier/Academic Press, Amsterdam, 2003.
|
| [2] |
K. Akô,
On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.
doi: 10.2969/jmsj/01310045. |
| [3] |
H. Amann,
Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295.
doi: 10.1007/BF01221152. |
| [4] |
H. Amann and M. G. Crandall,
On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779-790.
doi: 10.1512/iumj.1978.27.27050. |
| [5] |
J.-M. Bony,
Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Sér. A., 265 (1967), 333-336.
|
| [6] |
K. C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005. |
| [7] |
M. A. del Pino,
Positive solutions of a semilinear equation on a compact manifold, Nonlinear Analysis TMA, 22 (1994), 1423-1430.
doi: 10.1016/0362-546X(94)90121-X. |
| [8] |
P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Second edition, Birkhäuser/Springer Basel AG, Basel, 2013.
doi: 10.1007/978-3-0348-0387-8. |
| [9] |
A. Friedman, Partial Differential Equations, Dover Publications Inc., Mineola, New York, 1969/2008. |
| [10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Reprint of the 1998 edition, Springer-Verlag, New York Berlin Heidelberg Tokyo, 2001. |
| [11] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc., Academic Press, New York London, 1968.
|
| [12] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
| [13] |
M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J., 6 (1954), 207–229. https://projecteuclid.org/euclid.ojm/1200688553. |
| [14] |
T.-C. Ouyang,
On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^{p} = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527.
doi: 10.2307/2154124. |
| [15] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 3, Walter de Gruyter & Co., Berlin New York, 1996.
doi: 10.1515/9783110812411. |
| [16] |
K. Schmitt,
Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. TMA, 2 (1978), 263-309.
doi: 10.1016/0362-546X(78)90019-6. |
| [17] |
K. Taira,
The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.
doi: 10.1006/jdeq.1995.1151. |
| [18] |
K. Taira,
Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.
doi: 10.1007/BF02621868. |
| [19] |
K. Taira,
Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equations, 2 (1997), 509-534.
|
| [20] |
K. Taira,
Bifurcation theory for semilinear elliptic boundary value problems, Hiroshima Math. J., 28 (1998), 261-308.
doi: 10.32917/hmj/1206126761. |
| [21] |
K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer Monogr. Math., Second edition, Springer-Verlag, Berlin Heidelberg New York, 2014.
doi: 10.1007/978-3-662-43696-7. |
| [22] |
K. Taira, Analytic Semigroups and Semilinear Initial-Boundary Value Problems, London Mathematical Society Lecture Note Series, Vol. 434, Second edition, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316729755.
|
| [23] |
K. Taira, D. K. Palagachev and P. R. Popivanov,
A degenerate Neumann problem for quasilinear elliptic equations, Tokyo J. Math., 23 (2000), 227-234.
doi: 10.3836/tjm/1255958817. |
| [24] |
F. Tomi,
Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111 (1969), 350-366.
doi: 10.1007/BF01110746. |
| [25] |
G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics, Plenum Press, New York, 1987.
doi: 10.1007/978-1-4899-3614-1.
|
show all references
Dedicated to Professor Angelo Favini on the occasion of his 70th birthday
References:
| [1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Second edition, Elsevier/Academic Press, Amsterdam, 2003.
|
| [2] |
K. Akô,
On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.
doi: 10.2969/jmsj/01310045. |
| [3] |
H. Amann,
Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295.
doi: 10.1007/BF01221152. |
| [4] |
H. Amann and M. G. Crandall,
On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779-790.
doi: 10.1512/iumj.1978.27.27050. |
| [5] |
J.-M. Bony,
Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Sér. A., 265 (1967), 333-336.
|
| [6] |
K. C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005. |
| [7] |
M. A. del Pino,
Positive solutions of a semilinear equation on a compact manifold, Nonlinear Analysis TMA, 22 (1994), 1423-1430.
doi: 10.1016/0362-546X(94)90121-X. |
| [8] |
P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Second edition, Birkhäuser/Springer Basel AG, Basel, 2013.
doi: 10.1007/978-3-0348-0387-8. |
| [9] |
A. Friedman, Partial Differential Equations, Dover Publications Inc., Mineola, New York, 1969/2008. |
| [10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Reprint of the 1998 edition, Springer-Verlag, New York Berlin Heidelberg Tokyo, 2001. |
| [11] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc., Academic Press, New York London, 1968.
|
| [12] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
| [13] |
M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J., 6 (1954), 207–229. https://projecteuclid.org/euclid.ojm/1200688553. |
| [14] |
T.-C. Ouyang,
On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^{p} = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527.
doi: 10.2307/2154124. |
| [15] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 3, Walter de Gruyter & Co., Berlin New York, 1996.
doi: 10.1515/9783110812411. |
| [16] |
K. Schmitt,
Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. TMA, 2 (1978), 263-309.
doi: 10.1016/0362-546X(78)90019-6. |
| [17] |
K. Taira,
The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.
doi: 10.1006/jdeq.1995.1151. |
| [18] |
K. Taira,
Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.
doi: 10.1007/BF02621868. |
| [19] |
K. Taira,
Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equations, 2 (1997), 509-534.
|
| [20] |
K. Taira,
Bifurcation theory for semilinear elliptic boundary value problems, Hiroshima Math. J., 28 (1998), 261-308.
doi: 10.32917/hmj/1206126761. |
| [21] |
K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer Monogr. Math., Second edition, Springer-Verlag, Berlin Heidelberg New York, 2014.
doi: 10.1007/978-3-662-43696-7. |
| [22] |
K. Taira, Analytic Semigroups and Semilinear Initial-Boundary Value Problems, London Mathematical Society Lecture Note Series, Vol. 434, Second edition, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316729755.
|
| [23] |
K. Taira, D. K. Palagachev and P. R. Popivanov,
A degenerate Neumann problem for quasilinear elliptic equations, Tokyo J. Math., 23 (2000), 227-234.
doi: 10.3836/tjm/1255958817. |
| [24] |
F. Tomi,
Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111 (1969), 350-366.
doi: 10.1007/BF01110746. |
| [25] |
G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics, Plenum Press, New York, 1987.
doi: 10.1007/978-1-4899-3614-1.
|
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