American Institute of Mathematical Sciences

Advanced
January  2011, 10(1): 69-91. doi: 10.3934/cpaa.2011.10.69

A comparison principle for a Sobolev gradient semi-flow

Timothy Blass 1, , Rafael De La Llave 2, and  Enrico Valdinoci 3,

1. 

Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA Government
2. 

Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States
3. 

Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano, Italy

Received  January 2010 Revised  June 2010 Published  November 2010

We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} < u(x), A(x)u(x)>_{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration.
    We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow.
    We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.
Keywords: Comparison principle, fractional powers of elliptic operators., Sobolev gradient, semigroups of linear operators.
Mathematics Subject Classification: Primary: 35B50, 46N20; Secondary: 35J2.
Citation: Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
References:
[1]

S. Bochner, Diffusion equation and stochastic processes,, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368.  doi: doi:10.1073/pnas.35.7.368.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: doi:10.1002/cpa.20274.  Google Scholar

[3]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678.  doi: doi:10.1002/cpa.20093.  Google Scholar

[4]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25.   Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics. Springer-Verlag, (2001).   Google Scholar

[6]

M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications., Birkh\, (2006).   Google Scholar

[7]

T. Kato, Note on fractional powers of linear operators,, Proc. Japan Acad., 36 (1960), 94.  doi: doi:10.3792/pja/1195524082.  Google Scholar

[8]

O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Translated from the Russian by Scripta Technica, (1968).   Google Scholar

[9]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and their Applications, (1995).   Google Scholar

[10]

R. de la Llave and E. Valdinoci, A generalization of aubry-mather theory to partial differential equations and pseudo-differential equations,, Annales de l'Institut Henri Poincare C Non Linear Analysis, 26 (2009), 1309.   Google Scholar

[11]

C. Martínez Carracedo and M. Sanz Alix, "The Theory of Fractional Powers of Operators," volume 187 of North-Holland Mathematics Studies,, North-Holland Publishing Co., (2001).   Google Scholar

[12]

M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations,", World Scientific Publishing Co. Inc., (1998).   Google Scholar

[13]

J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265.   Google Scholar

[14]

J. Moser, Minimal solutions of variational problems on a torus,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 3 (1986), 229.   Google Scholar

[15]

J. Moser, A stability theorem for minimal foliations on a torus,, Ergodic Theory Dynam. Systems, (1988), 251.   Google Scholar

[16]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics,, Springer-Verlag, (1997).   Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences,, Springer-Verlag, (1983).   Google Scholar

[18]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall Inc., (1967).   Google Scholar

[19]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," volume 49 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997).   Google Scholar

[20]

M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,", Springer-Verlag, (2001).   Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).   Google Scholar

[22]

M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences., Springer-Verlag, (1996).   Google Scholar

[23]

M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences., Springer-Verlag, (1997).   Google Scholar

[24]

I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies., North-Holland Publishing Co., (2003).   Google Scholar

[25]

K. Yosida, "Functional Analysis,", Springer-Verlag, (1974).   Google Scholar

show all references

References:
[1]

S. Bochner, Diffusion equation and stochastic processes,, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368.  doi: doi:10.1073/pnas.35.7.368.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: doi:10.1002/cpa.20274.  Google Scholar

[3]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678.  doi: doi:10.1002/cpa.20093.  Google Scholar

[4]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25.   Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics. Springer-Verlag, (2001).   Google Scholar

[6]

M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications., Birkh\, (2006).   Google Scholar

[7]

T. Kato, Note on fractional powers of linear operators,, Proc. Japan Acad., 36 (1960), 94.  doi: doi:10.3792/pja/1195524082.  Google Scholar

[8]

O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Translated from the Russian by Scripta Technica, (1968).   Google Scholar

[9]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and their Applications, (1995).   Google Scholar

[10]

R. de la Llave and E. Valdinoci, A generalization of aubry-mather theory to partial differential equations and pseudo-differential equations,, Annales de l'Institut Henri Poincare C Non Linear Analysis, 26 (2009), 1309.   Google Scholar

[11]

C. Martínez Carracedo and M. Sanz Alix, "The Theory of Fractional Powers of Operators," volume 187 of North-Holland Mathematics Studies,, North-Holland Publishing Co., (2001).   Google Scholar

[12]

M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations,", World Scientific Publishing Co. Inc., (1998).   Google Scholar

[13]

J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265.   Google Scholar

[14]

J. Moser, Minimal solutions of variational problems on a torus,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 3 (1986), 229.   Google Scholar

[15]

J. Moser, A stability theorem for minimal foliations on a torus,, Ergodic Theory Dynam. Systems, (1988), 251.   Google Scholar

[16]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics,, Springer-Verlag, (1997).   Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences,, Springer-Verlag, (1983).   Google Scholar

[18]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall Inc., (1967).   Google Scholar

[19]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," volume 49 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997).   Google Scholar

[20]

M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,", Springer-Verlag, (2001).   Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).   Google Scholar

[22]

M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences., Springer-Verlag, (1996).   Google Scholar

[23]

M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences., Springer-Verlag, (1997).   Google Scholar

[24]

I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies., North-Holland Publishing Co., (2003).   Google Scholar

[25]

K. Yosida, "Functional Analysis,", Springer-Verlag, (1974).   Google Scholar

[1]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[2]

Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
[3]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335
[4]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581
[5]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[6]

V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481
[7]

Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43
[8]

Bertrand Lods, Mustapha Mokhtar-Kharroubi, Mohammed Sbihi. Spectral properties of general advection operators and weighted translation semigroups. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469
[9]

Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa. On the positive semigroups generated by Fleming-Viot type differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 323-340. doi: 10.3934/cpaa.2019017
[10]

Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
[11]

Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure & Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83
[12]

Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems & Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020
[13]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477
[14]

Giorgio Metafune, Chiara Spina, Cristian Tacelli. On a class of elliptic operators with unbounded diffusion coefficients. Evolution Equations & Control Theory, 2014, 3 (4) : 671-680. doi: 10.3934/eect.2014.3.671
[15]

Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure & Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[16]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
[17]

Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure & Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5
[18]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[19]

Thaís Jordão, Xingping Sun. General types of spherical mean operators and $K$-functionals of fractional orders. Communications on Pure & Applied Analysis, 2015, 14 (3) : 743-757. doi: 10.3934/cpaa.2015.14.743
[20]

Daliang Zhao, Yansheng Liu, Xiaodi Li. Controllability for a class of semilinear fractional evolution systems via resolvent operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 455-478. doi: 10.3934/cpaa.2019023

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences