Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers...

Let $L_{\alpha }^q(R^D),\alpha \>0,1\<q\<\infty$, denote the space of Bessel potentials $f=G_{\alpha }*g$, $g\in L^q$, with norm $\parallel f{\parallel }_{\alpha ,q}=\parallel g{\parallel }_q$. For $\alpha$ integer $L_{\alpha }^q$ can be identified with the Sobolev space $H^{\alpha ,q}$.One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space $H^{1;2}$, and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the positive cone in the space...

Compactness in the space $L^p(0,T;B)$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting...

We prove that if there is an open mapping from a subspace of $C_p\left(X\right)$ onto $C_p\left(Y\right)$, then $Y$ is a countable union of images of closed subspaces of finite powers of $X$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if $X$ and $Y$ are $t$-equivalent compact spaces, then $X$ and $Y$ have the same tightness, and that, assuming $2^{𝔱}>𝔠$, if $X$ and $Y$ are $t$-equivalent compact spaces and $X$ is sequential, then $Y$ is sequential.

We give necessary and sufficient conditions on the initial data such that the solutions of parabolic equations have a prescribed Sobolev regularity in time and space.

We study Toeplitz operators $T_a$ with radial symbols in weighted Bergman spaces $A_{\mu }^p$, 1 < p < ∞, on the disc. Using a decomposition of $A_{\mu }^p$ into finite-dimensional subspaces the operator $T_a$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_a$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_a$ for a satisfying an assumption on the positivity of certain indefinite...

We establish the topological relationship between compact Hausdorff spaces X and Y equivalent to the existence of a bound-2 isomorphism of the sup norm Banach spaces C(X) and C(Y).

Let $L^{\varphi }\left(X\right)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi$ taking only finite values (not necessarily convex) over a $\sigma$-finite atomless measure space. It is proved that the topological dual $L^{\varphi }{\left(X\right)}^{*}$ of $L^{\varphi }\left(X\right)$ can be represented in the form: $L^{\varphi }{\left(X\right)}^{*}=L^{\varphi }{\left(X\right)}_n^{\sim }\oplus L^{\varphi }{\left(X\right)}_s^{\sim }$, where $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ denote the order continuous dual and the singular dual of $L^{\varphi }\left(X\right)$ respectively. The spaces $L^{\varphi }{\left(X\right)}^{*}$, $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory...