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

A comparison principle for a Sobolev gradient semi-flow

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.
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]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[2]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[3]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[4]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[5]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[6]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[7]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[8]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[9]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[10]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[11]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[12]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[13]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[14]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[15]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[16]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[17]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[18]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[19]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[20]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (0)

[Back to Top]