Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

Armstrong, Scott N.

doi:10.1016/j.jde.2008.10.026

Cited by 57 publications

(82 citation statements)

References 33 publications

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“…We emphasize that we may also deduce P + λ(x,ω),Λ (D 2 w) ≥ 0 in the viscosity sense, and make other similar formal deductions rigorous, using [2], Lemma 3.2.…”

Section: Preliminariesmentioning

confidence: 71%

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Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Armstrong¹,

Smart²

2014

Ann. Probab.

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We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite dth moment, where d is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite pth moment, for every p < d, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.

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“…We emphasize that we may also deduce P + λ(x,ω),Λ (D 2 w) ≥ 0 in the viscosity sense, and make other similar formal deductions rigorous, using [2], Lemma 3.2.…”

Section: Preliminariesmentioning

confidence: 71%

“…[7,12]). We remark that, while it is not obvious-in fact, it is equivalent to the comparison principle-we have transitivity of inequalities in the viscosity sense (see [2], Lemma 3.2). For example, if V ∈ L and u, −v ∈ USC(V ) satisfy…”

Section: Preliminariesmentioning

confidence: 98%

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Armstrong¹,

Smart²

2014

Ann. Probab.

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“…We wish to emphasise that in [3], Armstrong defined Λ 2 = inf{λ > λ Hence the importance of any estimate on Λ 2 .…”

Section: Applications and Spectral Propertiesmentioning

confidence: 99%

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

Birindelli

Leoni

Pacella

2017

Journal de Mathématiques Pures et Appliquées

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Abstract. We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if u is a viscosity solution of a rotationally invariant equation of the form+ is the Pucci's sup-operator, plays the role of the linearized operator at u. In particular, we prove that if u is a solution in a radial bounded domain, if f is convex in u and if the principal eigenvalue of Lu (associated with positive eigenfunctions) in any half domain is nonnegative, then u is foliated Schwarz symmetric. We apply our symmetry results to obtain bounds on the spectrum and to deduce properties of possible nodal eigenfunctions for the operator M + . IntroductionThis paper studies symmetry properties of solutions of fully nonlinear equations related to spectral properties of what, improperly, will be called the linearized operator. The question we would like to answer is, which symmetry features of the domain and the operator are inherited by the viscosity solutions of the homogeneous Dirichlet problemwhere Ω ⊂ R n , n ≥ 2, is a bounded domain and F is a fully nonlinear uniformly elliptic operator. For the purpose of this introduction, let us emphasise its limit of application. Indeed, as it is well known by the experts, the moving plane method cannot be applied if the domain is not convex in the symmetry direction, say e.g. if Ω is an annulus, or if the nonlinear term f (x, u) does not have the right monotonicity in the x-variable (see e.g.[21] for 2010 Mathematics Subject Classification. 35J60.

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“…We observe the last condition in (3) means the reaction in the system dominates the absorption, so there is no conservation of mass in the timedependent version of (1). In [28] we studied the classification and non-existence of positive solutions of (1) in unbounded domains, as well as the a priori bounds and existence results which can be inferred in smooth bounded domains.…”

Section: Introductionmentioning

confidence: 99%

Stationary States of Reaction-Diffusion and Schrödinger Systems with Inhomogeneous or Controlled Diffusion

Montaru¹,

Sirakov²

2016

SIAM J. Math. Anal.

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Abstract. We obtain classification, solvability and nonexistence theorems for positive stationary states of reaction-diffusion and Schrödinger systems involving a balance between repulsive and attractive terms. This class of systems contains PDE arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions. We show, with different proofs, that the results obtained in [ARMA, 213 (2014), 129-169] for models with homogeneous diffusion are valid for general heterogeneous media, and even for controlled inhomogeneous diffusions.

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Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

Cited by 57 publications

References 33 publications

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

Stationary States of Reaction-Diffusion and Schrödinger Systems with Inhomogeneous or Controlled Diffusion

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