We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In this classical case it is known that, at least in a suitable weak sense, solutions of non-homogeneous Dirichlet problem are unique and tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that such results are false for equations driven by a wide class of nonlocal fractional operators, extending previous findings for some models of the fractional Laplacian operator. Actually, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function. Contents

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) and their embeddings, for $$s \in (0,1]$$ s ∈ ( 0 , 1 ] and $$p\ge 1$$ p ≥ 1 . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For $$s\,p < n$$ s p < n or $$s = p = n = 1$$ s = p = n = 1 we show that $$\mathcal {D}^{s,p}(\mathbb {R}^n)$$ D s , p ( R n ) is isomorphic to a suitable function space, whereas for $$s\,p \ge n$$ s p ≥ n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.

We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation (−∆) s u + V u = f posed in bounded domain Ω ⊂ R n with zero outside conditions. We consider general nonnegative potentials V ∈ L 1 loc (Ω) and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f . Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration since it leads to so-called flat solutions. We comment on related literature.

We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case. Keywords: Lithium-ion battery cell, multiscale mathematical model, Green operators, fixed point theory, Browder-Minty existence results, super and sub solutions 2010 MSC: 35M10, 35Q60, 35C15, 35B50, 35B60• The electric potential φ s = φ s (x, t) in the electrodes.• The electric potential measured by a reference Lithium electrode in the electrolyte, ϕ e = ϕ e (x, t).• The temperature T (t) in the cell.The system of equations is given by (5)-(9) below.