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We study embeddings of fractional Sobolev spaces defined on metric-measure spaces. Various results about continuous and compact embeddings are proven. Many theorems are illustrated by a number of examples.
We study a characterization of the precompactness of sets in variable exponent Morrey spaces on bounded metric measure spaces. Totally bounded sets are characterized from several points of view for the case of variable exponent Morrey spaces over metric measure spaces. This characterization is new in the case of constant exponents.
In this note we show that, in the case of bounded sets in metric spaces with some additional structure, the boundedness of a family of Lebesgue p-summable functions follow from a certain uniform limit norm condition. As a byproduct, the well known Riesz-Kolmogorov compactness theorem can be formulated only with one condition.
In this note we correct a misprint in the formulation of Theorem 4.5 of paper Górka (2016). We give the correct formulation and provide a friendlier proof.
In this paper we study totally bounded sets in Banach function spaces (BFS), from which we characterize compact sets (via Hausdorff criterion) in some non-standard function spaces which fall under the umbrella of BFS. We obtain a Riesz–Kolmogorov compactness theorem for the grand variable exponent Lebesgue spaces.
We study Trudinger-Moser type inequalities on the entire metric measure spaces Moreover, we give the necessary and sufficient conditions under which the Trudinger-Mose inequality holds.
We study the motion of the so-called bent rectangles by the singular weighted mean curvature. We are interested in the curves which can be rendered as graphs over a smooth one-dimensional reference manifold. We establish a sufficient condition for that. Once we deal with graphs we can have the tools of the viscosity theory available, like the Compa...
We study relatively compact sets in variable Lebesgue spaces. The full characterization of such sets is given in the case of variable Lebesgue space on metric measure spaces. The paper contains a detailed discussion of relatively compact sets in variable Lebesgue space on the Euclidean space. Moreover, some applications of the theorems are given.
We present an existence and uniqueness result for ordinary differential equations based upon the use of weighted spaces. We deal with singular and discontinuous right-hand-sides of the equation. Mathematics Subject Classification (2000)Primary 34A12-Secondary 34A34 KeywordsOrdinary differential equation-Existence-Uniqueness-Weighted spaces
We consider the weighted mean curvature flow with a driving term in the plane. For anisotropy functions this evolution problem degenerates to a first order Hamilton-Jacobi equation with a free boundary. The resulting problem may be written as a Hamilton-Jacobi equation with a spatially non-local and discontinuous Hamiltonian. We prove existence and...