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A certain example for a De Giorgi conjecture
조성원 충청수학회 2014 충청수학회지 Vol.27 No.4
In this paper, we illustrate a counter example forthe converse of a certain conjecture proposed by De Giorgi. De Giorgi suggested a series of conjectures, in which a certain integral conditionfor singularity or degeneracyof an elliptic operator is satisfied, the solutions are continuous. We construct some singular elliptic operators and solutions such that the integral condition does not hold,but the solutions are continuous.
A CERTAIN EXAMPLE FOR A DE GIORGI CONJECTURE
Cho, Sungwon Chungcheong Mathematical Society 2014 충청수학회지 Vol.27 No.4
In this paper, we illustrate a counter example for the converse of a certain conjecture proposed by De Giorgi. De Giorgi suggested a series of conjectures, in which a certain integral condition for singularity or degeneracy of an elliptic operator is satisfied, the solutions are continuous. We construct some singular elliptic operators and solutions such that the integral condition does not hold, but the solutions are continuous.
A Priori Boundary Estimations for an Elliptic Operator
Cho, Sungwon The Basic Science Institute Chosun University 2014 조선자연과학논문집 Vol.7 No.4
In this article, we consider a singular and a degenerate elliptic operators in a divergence form. The singularities exist on a part of boundary, and comparable to the logarithmic distance function or its inverse. If we assume that the operator can be treated in a pointwise sense than distribution sense, with this operator we obtain a priori Harnack continuity near the boundary. In the proof we transform the singular elliptic operator to uniformly bounded elliptic operator with unbounded first order terms. We study this type of estimations considering a De Giorgi conjecture. In his conjecture, he proposed a certain ellipticity condition to guarantee a continuity of a solution.
A Priori Boundary Estimations for an Elliptic Operator
조성원 조선대학교 기초과학연구원 2014 조선자연과학논문집 Vol.7 No.4
In this article, we consider a singular and a degenerate elliptic operators in a divergence form. The singularities exist on a part of boundary, and comparable to the logarithmic distance function or its inverse. If we assume that the operator can be treated in a pointwise sense than distribution sense, with this operator we obtain a priori Harnack continuity near the boundary. In the proof we transform the singular elliptic operator to uniformly bounded elliptic operator with unbounded first order terms. We study this type of estimations considering a De Giorgi conjecture. In his conjecture, he proposed a certain ellipticity condition to guarantee a continuity of a solution.