Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities

Feehan, Paul M. N.; Pop, Camelia A.

doi:10.1016/j.anihpc.2016.07.005

Cited by 17 publications

(20 citation statements)

References 40 publications

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“…However, the proof of even the weak maximum principle for a weakly A-subharmonic function in W 1,2 (O) is neither as direct nor as intuitive as that presented here, since it relies on a certain weighted Sobolev inequality (see [19,Theorem 8.8]). Moreover, an analogue of the classical proof of the strong maximum principle (see [27,Theorem 8.19]) would require an analogue of the classical weak Harnack inequality (see [27,Theorem 8.18]) valid for the boundary-degenerate elliptic operator, A, written in divergence form (see [20,Theorem 1.23] for such a result in the case of the Heston operator [30]). The proof of our Harnack inequality for the boundary-degenerate elliptic Heston operator, developed with C. Pop, is lengthy and difficult, relying on a Moser iteration argument, the abstract John-Nirenberg inequality due to Bombieri and Giusti [4,Theorem 4], Poincaré and Sobolev inequalities for weighted Sobolev spaces [20, §2], and delicate choices of test functions.…”

Section: 3mentioning

confidence: 99%

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

Feehan

2020

Trans. Amer. Math. Soc.

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Abstract. We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, Au := − tr(aD 2 u) − b, Du + cu, with partial Dirichlet boundary conditions. The coefficient, a(x), is assumed to vanish along a non-empty open subset, ∂0O, called the degenerate boundary portion, of the boundary, ∂O, of the domain O ⊂ R d , while a(x) is non-zero at any point of the non-degenerate boundary portion,loc (O), is C 1 up to ∂0O and has a strict local maximum at a point in ∂0O, we show that u can be perturbed, by the addition of a suitable function, to a strictly A-subharmonic function v = u + w having a local maximum in the interior of O. Consequently, we obtain strong and weak maximum principles for A-subharmonic functions in C 2 (O) and W 2,dloc (O) which are C 1 up to ∂0O. Points in ∂0O play the same role as those in the interior of the domain, O, and only the non-degenerate boundary portion, ∂1O, is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W 2,d loc (O) to a unilateral obstacle problem defined by A, again where only the non-degenerate boundary portion, ∂1O, is required for boundary comparisons. Our results extend those in [12,15,19,18], where tr(aD 2 u) is in addition assumed to be continuous up to and vanish along ∂0O in order to yield comparable maximum principles for A-subharmonic functions in C 2 (O), while the results developed here for A-subharmonic functions in W

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Section: 3mentioning

confidence: 99%

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

Feehan

2020

Trans. Amer. Math. Soc.

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“…In [11] it is shown that this natural boundary condition can be understood as imposing a regularity requirement along these boundary components. Similar problems without boundary conditions on suitable portions of the boundary have been studied in [8,34,21,20,22,19], among others.…”

Section: Remark 23 (Boundary Condition Alongmentioning

confidence: 99%

“…, such that for all 4r 2 < t < T − 4r 2 and for all p ∈ ∂P we have that sup Finally, we remark that the class of processes described by generalized Kimura operators appear not only in population genetics, but they are also encountered in the study of superprocesses, [1,2], of Fleming-Viot processes in population dynamics, [3,4,5,6], and are closely related to the linearization of the porous medium equation, [8,34], to affine models for interest rates, [7,10], and to stochastic volatility models in mathematical finance, [31,20,21,22,23,24,25].…”

Section: Theorem 12 (Boundary Regularity)mentioning

confidence: 99%

Boundary Estimates for a Degenerate Parabolic Equation with Partial Dirichlet Boundary Conditions

Epstein

Pop

2017

J Geom Anal

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ABSTRACT. We study the boundary regularity properties and derive pointwise a priori supremum estimates of weak solutions and their derivatives in terms of suitable weighted L 2 -norms for a class of degenerate parabolic equations that satisfy homogeneous Dirichlet boundary conditions on certain portions of the boundary. Such equations arise in population genetics in the study of models for the evolution of gene frequencies. Among the applications of our results is the description of the structure of the transition probabilities and of the hitting distributions of the underlying gene frequencies process.

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“…However, in this setting, the boundary condition is not mixed, since there is no Dirichlet condition along any portion of ∂O, only the condition (1.8) implied by the regularity of u up to ∂O. By extending the methods of Koch [26] and restricting to the case where A is the Heston operator (1.24) and d = 2, Daskalopoulos, Pop, and the author succeeded in proving existence, uniqueness, and regularity os solutions to the equation (1.1) and obstacle problem (1.3) with partial Dirichlet boundary condition (1.2) by solving the associated variational equation and inequality for solutions, u, in weighted Sobolev spaces [8]; appealing to [13] to obtain uniqueness; proving that the solutions are continuous up to the boundary [15] using a Moser iteration technique; proving Schauder regularity when f ∈ C ∞ 0 (O) ∩ C b (O) using a variational method [16]; proving the expected Schauder regularity, u ∈ C 2+α…”

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confidence: 99%

“…However, the Perron methods which we use in this article, and which we outline in §1.7, provide a more direct and elegant path to the desired results, even though they do not establish continuity of the solution at the corner points, although such continuity properties are proved by Pop and the author by a variational method in [15] for the case of the Heston operator (1.24). It is important to note that the Perron methods developed in this article are analogues of their classical counterpart in [20,Chapters 2 and 6] for the existence of smooth solutions to a Dirichlet problem for a linear, second-order, strictly elliptic operator.…”

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confidence: 99%

A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps

Feehan

2017

Journal of Differential Equations

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Abstract. We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton [9] in their study of the porous medium equation or the degeneracy of the Heston operator [22] in mathematical finance. Existence of a solution to the partial Dirichlet problem on a half-ball, where the operator becomes degenerate on the flat boundary and a Dirichlet condition is only imposed on the spherical boundary, provides the key additional ingredient required for our Perron method. Surprisingly, proving existence of a solution to this partial Dirichlet problem with "mixed" boundary conditions on a half-ball is more challenging than one might expect. Due to the difficulty in developing a global Schauder estimate and due to compatibility conditions arising where the "degenerate" and "non-degenerate boundaries" touch, one cannot directly apply the continuity or approximate solution methods. However, in dimension two, there is a holomorphic map from the half-disk onto the infinite strip in the complex plane and one can extend this definition to higher dimensions to give a diffeomorphism from the half-ball onto the infinite "slab". The solution to the partial Dirichlet problem on the half-ball can thus be converted to a partial Dirichlet problem on the slab, albeit for an operator which now has exponentially growing coefficients. The required Schauder regularity theory and existence of a solution to the partial Dirichlet problem on the slab can nevertheless be obtained using previous work of the author and C. Pop [17]. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates previously developed by the author [13].

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Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities

Cited by 17 publications

References 40 publications

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

Boundary Estimates for a Degenerate Parabolic Equation with Partial Dirichlet Boundary Conditions

A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps

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