Weak solutions with a dense set of discontinuities

Weak differentiability of solutions to SDEs with semi-monotone drifts

‎In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift‎. ‎To prove this formula‎, ‎we construct a sequence of SDEs with globally Lipschitz drifts and show that the $p$-moments of their Malliavin derivatives are uniformly bounded‎.

Singular Radial Entire Solutions and Weak Solutions with Prescribed Singular Set for a Biharmonic Equation

Positive singular radial entire solutions of a biharmonic equation with subcritical exponent are obtained via the entire radial solutions of the equation with supercritical exponent and the Kelvin transformation. The expansions of such singular radial solutions at the singular point 0 are presented. Using these singular radial entire solutions, we construct solutions with a prescribed singular ...

Weak Compactness of the Set of ε-Extensions

Existence of Positive Weak Solutions with a Prescribed Singular Set of Semilinear Elliptic Equations

In this paper, we consider the problem of the existence of non-negative weak solution u of A u + u P = O i n ~ u = 0 on Of~ n n + 2 ~ / n 1 having a given closed set S as its singular set. We prove that when < p < _ _ and n 2 n 4 + 2 nv/'n-L--1 S is a closed subset of f2, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal sca...

Using necessarily weak efficient solutions for solving a biobjective transportation problem with fuzzy objective functions coefficients

This paper considers a biobjective transportation problem with various fuzzy objective functions coefficients. Fuzzy coefficients can be of different types such as triangular, trapezoidal, (semi) $L-R$, or flat (semi) $L-R$ fuzzy numbers. First, we convert the problem to a parametric interval biobjective transportation problem using $gamma$-cuts of fuzzy coefficients. Then, we consider a fix $g...