Concentrated terms and varying domains in elliptic equations: Lipschitz case

G.S. Arag\~ao; S. M Bruschi

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Concentrated terms and varying domains in elliptic equations: Lipschitz case

G.S. Aragão, S. M Bruschi

Analysis of PDEs · Submitted 18 Dec 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.05 · mentions code or data availability 0.01.

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In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain.	contribution 0.86	0.52	0.05	asserted 0.91
s1	We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary.	result 0.99	0.95	0.87	asserted 1.00
s2	Indeed, we show the upper semicontinuity of this family of solutions.	result 0.98	0.94	0.12	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.

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The dynamics of the flow generated by a nonlinear parabolic problem posed on a $C^2$ domain $Ω$ in $^2$, when some reaction and potential terms are concentrated in a neighborhood of the boundary and the “inner boundary” of this neighborhood presents a highly oscillatory behavior, was studied in [ref] where the continuity of the family of attractors was proved.

Continuity of attractors for a nonlinear parabolic problem with terms concentrating in the boundary · Introduction · 0.87

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