On a singular quasilinear anisotropic elliptic boundary value problem

Choi, Y. S.; McKenna, P. J.

doi:10.1090/s0002-9947-1995-1277103-8

Cited by 25 publications

(41 citation statements)

References 10 publications

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“…This allows us to obtain L ∞ bounds on the solutions to this class of equations, using the method of upper and lower solutions. The proof is similar to that of lemma 1 in [2], and hence will be skipped.…”

Section: A Comparison Lemmamentioning

confidence: 98%

“…By the main theorem in [5] (which is restated in theorem 2 in [2]), a unique positive solution ψ of (1.…”

Section: Theorem 3 Let All the Assumptions In Theorem 1 Hold Exceptmentioning

confidence: 99%

“…The purpose of this paper is to answer two of the questions left open in [2]. Recall that that paper began the study of the boundary value problem u a u xx + u b u yy + p(x) = 0, (1.1) u| ∂Ω = 0, (1.2) in a smooth bounded convex domain Ω in R 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Recall that that paper began the study of the boundary value problem u a u xx + u b u yy + p(x) = 0, (1.1) u| ∂Ω = 0, (1.2) in a smooth bounded convex domain Ω in R 2 . Here a and b are non-negative, but are not equal, and the function p was positive and continuous on the closure of Ω.…”

Section: Introductionmentioning

confidence: 99%

“…The solution has to be extracted by a compactness argument. The proof in [2] made use of an interior estimate that is valid only in two dimensions. More seriously, the convex domain assumption is crucial in the construction of an upper solution satisfying the boundary condition.…”

Section: Introductionmentioning

confidence: 99%

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A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II

Choi

McKenna

1998

Trans. Amer. Math. Soc.

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Abstract. Let Ω ⊂ R N with N ≥ 2. We consider the equationswith a 1 ≥ a 2 ≥ .... ≥ a N ≥ 0 and a 1 > a N . We show that if Ω is a convex bounded region in R N , there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in R 2 are also given.

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Section: A Comparison Lemmamentioning

confidence: 98%

“…By the main theorem in [5] (which is restated in theorem 2 in [2]), a unique positive solution ψ of (1.…”

Section: Theorem 3 Let All the Assumptions In Theorem 1 Hold Exceptmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II

Choi

McKenna

1998

Trans. Amer. Math. Soc.

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A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks

Čanić

Keyfitz

Kim

2001

Comm Pure Appl Math

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We prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. In weak regular reflection, the state immediately behind the reflected shock is supersonic and constant. The flow becomes subsonic further downstream; the equation in self-similar coordinates is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation. Using the Rankine-Hugoniot conditions along the reflected shock, we derive an evolution equation for the transonic shock, and an oblique derivative boundary condition at the unknown shock position. By regularizing the degenerate problem, we construct uniform bounds; we apply local compactness arguments to extract a limit that solves the problem. The solution is smooth in the interior and continuous up to the degenerate boundary.This work completes a stage in our program to construct self-similar solutions of two-dimensional Riemann problems. In a series of papers, we developed techniques for solving the degenerate elliptic equations that arise in self-similar reductions of hyperbolic conservation laws. In other papers, especially in joint work with Gary Lieberman, we developed techniques for solving free boundary problems of the type that arise from Rankine-Hugoniot relations. For the first time, in this paper, we combine these approaches and show that they are compatible. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection, including Mach and von Neumann reflection in the UTSD equation, and the analogous problems for the unsteady full potential equation.

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