The Experts below are selected from a list of 7941 Experts worldwide ranked by ideXlab platform
Naoyuki Ichihara - One of the best experts on this subject based on the ideXlab platform.
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Sharp estimates of the generalized Principal Eigenvalue for superlinear viscous Hamilton-Jacobi equations with inward drift
2019Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for viscous Hamilton-Jacobi equations having superlinear Hamiltonian, inward-pointing drift, and positive potential which vanishes at infinity. Assuming some radial symmetry of the drift and the potential outside a ball, we establish sharp estimates of the generalized Principal Eigenvalue with respect to a perturbation of the potential.
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Qualitative properties of generalized Principal Eigenvalues for superquadratic viscous Hamilton-Jacobi equations
Nonlinear Differential Equations and Applications, 2016Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m > 2. We prove that the generalized Principal Eigenvalue of the equation converges to a constant as m → ∞, and that the limit coincides with the generalized Principal Eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 < m < ∞, and the limiting case m = ∞.
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Qualitative properties of generalized Principal Eigenvalues for superquadratic viscous Hamilton-Jacobi equations
Nonlinear Differential Equations and Applications NoDEA, 2016Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for superquadratic viscous Hamilton–Jacobi equations with exponent $$m>2$$ . We prove that the generalized Principal Eigenvalue of the equation converges to a constant as $$m\rightarrow \infty $$ , and that the limit coincides with the generalized Principal Eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to $$m=2$$ , $$2
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Qualitative properties of generalized Principal Eigenvalues for superquadratic viscous Hamilton-Jacobi equations
arXiv: Analysis of PDEs, 2016Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m \textgreater{} 2. We prove that the generalized Principal Eigenvalue of the equation converges to a constant as m $\rightarrow$ $\infty$, and that the limit coincides with the generalized Principal Eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 \textless{} m \textless{} $\infty$, and the limiting case m = $\infty$.
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the generalized Principal Eigenvalue for hamilton jacobi bellman equations of ergodic type
Annales De L Institut Henri Poincare-analyse Non Lineaire, 2015Co-Authors: Naoyuki IchiharaAbstract:Abstract This paper is concerned with the generalized Principal Eigenvalue for Hamilton–Jacobi–Bellman (HJB) equations arising in a class of stochastic ergodic control. We give a necessary and sufficient condition so that the generalized Principal Eigenvalue of an HJB equation coincides with the optimal value of the corresponding ergodic control problem. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function.
Yuan Lou - One of the best experts on this subject based on the ideXlab platform.
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the generalised Principal Eigenvalue of time periodic nonlocal dispersal operators and applications
Journal of Differential Equations, 2020Co-Authors: Yuan Lou, Feiying YangAbstract:Abstract This paper is mainly concerned with the generalised Principal Eigenvalue for time-periodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised Principal Eigenvalue. We further investigate the dependence of the generalised Principal Eigenvalue on the frequency of the periodic oscillation, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency of the periodic oscillation, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.
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the generalised Principal Eigenvalue of time periodic nonlocal dispersal operators and applications
arXiv: Analysis of PDEs, 2019Co-Authors: Yuan Lou, Feiying YangAbstract:This paper is mainly concerned with the generalised Principal Eigenvalue for time-periodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised Principal Eigenvalue. We further investigate the dependence of the generalised Principal Eigenvalue on the frequency, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.
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monotonicity of the Principal Eigenvalue for a linear time periodic parabolic operator
arXiv: Analysis of PDEs, 2019Co-Authors: Shuang Liu, Yuan Lou, Rui Peng, Maolin ZhouAbstract:We investigate the effect of frequency on the Principal Eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the Principal Eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow and Polacik [2001 J. Math. Biol.].
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asymptotic behavior of the Principal Eigenvalue for cooperative elliptic systems and applications
Journal of Dynamics and Differential Equations, 2016Co-Authors: Kingyeung Lam, Yuan LouAbstract:The asymptotic behavior of the Principal Eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reaction-diffusion system converges uniformly to the equilibrium of the corresponding kinetic system.
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Principal Eigenvalue for an elliptic problem with indefinite weight on cylindrical domains
Mathematical Biosciences and Engineering, 2008Co-Authors: Chiuyen Kao, Yuan Lou, Eiji YanagidaAbstract:This paper is concerned with an indefinite weight linear Eigenvalue problem in cylindrical domains. We investigate the minimization of the positive Principal Eigenvalue under the constraint that the weight is bounded by a positive and a negative constant and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. Both our analysis and numerical simulations for rectangular domains indicate that there exists a threshold value such that if the total weight is below this threshold value, then the optimal favorable region is a circular-type domain at one of the four corners, and a strip at the one end with shorter edge otherwise.
Henri Berestycki - One of the best experts on this subject based on the ideXlab platform.
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on the definition and the properties of the Principal Eigenvalue of some nonlocal operators
Journal of Functional Analysis, 2016Co-Authors: Henri Berestycki, Jerome CovilleAbstract:In this article we study some spectral properties of the linear operator $\mathcal{L}_{\Omega}+a$ defined on the space $C(\bar\Omega)$ by : $$ \mathcal{L}_{\Omega}[\varphi] +a\varphi:=\int_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ where $\Omega\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised Principal Eigenvalue $\lambda_p(\mathcal{L}_{\Omega}+a)$ defined by $$\lambda_p(\mathcal{L}_{\Omega}+a):= \sup\{\lambda \in \mathbb{R} \,|\, \exists \varphi \in C(\bar \Omega), \varphi>0, \textit{ such that }\; \mathcal{L}_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \; \text{ in }\;\Omega\}. $$ We establish some new properties of this generalised Principal Eigenvalue $\lambda_p$. Namely, we prove the equivalence of different definitions of the Principal Eigenvalue. We also study the behaviour of $\lambda_p(\mathcal{L}_{\Omega}+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $\lambda_{p} \left(\mathcal{L}_{\sigma,m,\Omega} -\frac{1}{\sigma^m}+a\right)$ where $\mathcal{L}_{\sigma,m,\Omega}$ is defined by $\displaystyle{\mathcal{L}_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\, dy}$. In particular, we prove that $$\lim_{\sigma\to 0}\lambda_p\left(\mathcal{L}_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda_1\left(\frac{D_2(J)}{2N}\Delta +a\right),$$ where $D_2(J):=\int_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $\lambda_1$ denotes the Dirichlet Principal Eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\varphi_{p,\sigma}$.
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On the definition and the properties of the Principal Eigenvalue of some nonlocal operators
Journal of Functional Analysis, 2016Co-Authors: Henri Berestycki, Jerome CovilleAbstract:In this article we study some spectral properties of the linear operator L-Omega + a defined on the space C((Omega) over bar) by: L-Omega[phi] + a phi := integral(Omega) K(x, y)phi(y) dy + a(x)phi(x) where Omega C R-N is a domain, possibly unbounded, a is a continuous bounded function and K is a continuous, non-negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised Principal Eigenvalue lambda(p)(L-Omega + a) defined by lambda(p)(L-Omega + a) := sup{lambda is an element of R vertical bar there exists phi is an element of C((Omega) over bar), phi > 0, such that L-Omega[phi] + a phi + lambda phi 0) lambda(p) (L-sigma,L-m,L-Omega - 1/sigma(2) + a) = lambda(1) (D-2(J)/2N Delta+a), where D-2(J) := integral(RN) J(z)vertical bar z vertical bar(2) dz and lambda(1) enotes the Dirichlet Principal Eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction phi(p,sigma).
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on the definition and the properties of the Principal Eigenvalue of some nonlocal operators
arXiv: Analysis of PDEs, 2015Co-Authors: Henri Berestycki, Jerome CovilleAbstract:In this article we study some spectral properties of the linear operator $\mathcal{L}\_{\Omega}+a$ defined on the space $C(\bar\Omega)$ by :$$ \mathcal{L}\_{\Omega}[\varphi] +a\varphi:=\int\_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ where $\Omega\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised Principal Eigenvalue $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ defined by $$\lambda\_p(\mathcal{L}\_{\Omega}+a):= \sup\{\lambda \in \mathbb{R} \,|\, \exists \varphi \in C(\bar \Omega), \varphi\textgreater{}0, \textit{such that}\, \mathcal{L}\_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \, \text{in}\;\Omega\}. $$ We establish some new properties of this generalised Principal Eigenvalue $\lambda\_p$. Namely, we prove the equivalence of different definitions of the Principal Eigenvalue. We also study the behaviour of $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $\lambda\_{p} \left(\mathcal{L}\_{\sigma,m,\Omega} -\frac{1}{\sigma^m}+a\right)$ where $\mathcal{L}\_{\sigma,m,\Omega}$ is defined by $\displaystyle{\mathcal{L}\_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int\_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\, dy}$. In particular, we prove that $$\lim\_{\sigma\to 0}\lambda\_p\left(\mathcal{L}\_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda\_1\left(\frac{D\_2(J)}{2N}\Delta +a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $\lambda\_1$ denotes the Dirichlet Principal Eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\varphi\_{p,\sigma}$.
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generalizations and properties of the Principal Eigenvalue of elliptic operators in unbounded domains
Communications on Pure and Applied Mathematics, 2015Co-Authors: Henri Berestycki, Luca RossiAbstract:Using three different notions of the generalized Principal Eigenvalue of linear second-order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these Principal Eigenvalues, their simplicity, and several other properties are further discussed. © 2015 Wiley Periodicals, Inc.
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Maximum Principle and generalized Principal Eigenvalue for degenerate elliptic operators
Journal de Mathématiques Pures et Appliquées, 2015Co-Authors: Henri Berestycki, Italo Capuzzo Dolcetta, Alessio Porretta, Luca RossiAbstract:We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized Principal Eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized Principal Eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators. We further discuss the relations between this notion and other natural generalizations of the classical notion of Principal Eigenvalue, some of which have been previously introduced for particular classes of operators.
Emmanuel Chasseigne - One of the best experts on this subject based on the ideXlab platform.
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Sharp estimates of the generalized Principal Eigenvalue for superlinear viscous Hamilton-Jacobi equations with inward drift
2019Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for viscous Hamilton-Jacobi equations having superlinear Hamiltonian, inward-pointing drift, and positive potential which vanishes at infinity. Assuming some radial symmetry of the drift and the potential outside a ball, we establish sharp estimates of the generalized Principal Eigenvalue with respect to a perturbation of the potential.
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Qualitative properties of generalized Principal Eigenvalues for superquadratic viscous Hamilton-Jacobi equations
Nonlinear Differential Equations and Applications, 2016Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m > 2. We prove that the generalized Principal Eigenvalue of the equation converges to a constant as m → ∞, and that the limit coincides with the generalized Principal Eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 < m < ∞, and the limiting case m = ∞.
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Qualitative properties of generalized Principal Eigenvalues for superquadratic viscous Hamilton-Jacobi equations
Nonlinear Differential Equations and Applications NoDEA, 2016Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for superquadratic viscous Hamilton–Jacobi equations with exponent $$m>2$$ . We prove that the generalized Principal Eigenvalue of the equation converges to a constant as $$m\rightarrow \infty $$ , and that the limit coincides with the generalized Principal Eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to $$m=2$$ , $$2
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Qualitative properties of generalized Principal Eigenvalues for superquadratic viscous Hamilton-Jacobi equations
arXiv: Analysis of PDEs, 2016Co-Authors: Emmanuel Chasseigne, Naoyuki IchiharaAbstract:This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m \textgreater{} 2. We prove that the generalized Principal Eigenvalue of the equation converges to a constant as m $\rightarrow$ $\infty$, and that the limit coincides with the generalized Principal Eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized Principal Eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 \textless{} m \textless{} $\infty$, and the limiting case m = $\infty$.
Luca Rossi - One of the best experts on this subject based on the ideXlab platform.
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generalizations and properties of the Principal Eigenvalue of elliptic operators in unbounded domains
Communications on Pure and Applied Mathematics, 2015Co-Authors: Henri Berestycki, Luca RossiAbstract:Using three different notions of the generalized Principal Eigenvalue of linear second-order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these Principal Eigenvalues, their simplicity, and several other properties are further discussed. © 2015 Wiley Periodicals, Inc.
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Maximum Principle and generalized Principal Eigenvalue for degenerate elliptic operators
Journal de Mathématiques Pures et Appliquées, 2015Co-Authors: Henri Berestycki, Italo Capuzzo Dolcetta, Alessio Porretta, Luca RossiAbstract:We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized Principal Eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized Principal Eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators. We further discuss the relations between this notion and other natural generalizations of the classical notion of Principal Eigenvalue, some of which have been previously introduced for particular classes of operators.
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Maximum Principle and generalized Principal Eigenvalue for degenerate elliptic operators
arXiv: Analysis of PDEs, 2013Co-Authors: Henri Berestycki, Italo Capuzzo Dolcetta, Alessio Porretta, Luca RossiAbstract:We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized Principal Eigenvalue. Here, maximum principle refers to the non-positivity of viscosity subsolutions of the Dirichlet problem. This characterization is derived in terms of a new notion of generalized Principal Eigenvalue, which is needed because of the possible degeneracy of the operator, admitted in full generality. We further discuss the relations between this notion and other natural generalizations of the classical notion of Principal Eigenvalue, some of which had already been used in the literature for particular classes of operators.
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generalizations and properties of the Principal Eigenvalue of elliptic operators in unbounded domains
arXiv: Analysis of PDEs, 2010Co-Authors: Henri Berestycki, Luca RossiAbstract:Using three different notions of generalized Principal Eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions satisfying Dirichlet boundary conditions. Relations between these Principal Eigenvalues, their simplicity and several other properties are further discussed.
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on the Principal Eigenvalue of elliptic operators in r n and applications
Journal of the European Mathematical Society, 2006Co-Authors: Henri Berestychi, Luca RossiAbstract:Two generalizations of the notion of Principal Eigenvalue for elliptic operators in $\R^N$ are examined in this paper. We prove several results comparing these two Eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and ``limit periodic'' operators. These results apply to questions of existence and uniqueness for some semi-linear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures.
