In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system, we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularising terms.

In this paper, we obtain gradient estimates of the positive solutions to weighted p-Laplacian type equations with a gradient-dependent nonlinearity of the form 0.1

$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$

$\Omega \subseteq {\open R}^N$

$N\ges 2$

$m,q\in [0,\infty )$

$1<p\les N+\sigma $

$q>\max \{p-m-1,\sigma +\tau -1\}$

$ \vert \nabla u \vert $

$ \vert \nabla u \vert $

Let 1 < p < +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type

$-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$

We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.

We study the problem −∆pu = f(x, u) + t in Ω with Neumann boundary condition |∇u|p−2(∂u/∂v) = 0 on ∂Ω. There exists a t0 ∈ ℝ such that for t > t0 there is no solution. If t ≤ t0, there is at least a minimal solution, and for t < t0 there are at least two distinct solutions. We use the sub–supersolution method, a priori estimates and degree theory.

We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in timestrong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence)of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$ . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

We semi-discretize in space a time-dependent Navier-Stokes systemon a three-dimensional polyhedron by finite-elements schemesdefined on two grids. In the first step, the fully non-linearproblem is semi-discretized on a coarse grid, with mesh-size H.In the second step, the problem is linearized by substitutinginto the non-linear term, the velocity u H computed at stepone, and the linearized problem is semi-discretized on a finegrid with mesh-size h. This approach is motivated by the factthat, on a convex polyhedron and under adequate assumptions on thedata, the contribution of u H to the error analysis ismeasured in the L 2 norm in space and time, and thus, for thelowest-degree elements, is of the order of H 2. Hence, an errorof the order of h can be recovered at the second step, providedh = H 2.