Riccardo Durastanti

Bio: Riccardo Durastanti is an academic researcher from Sapienza University of Rome. The author has contributed to research in topics: Nabla symbol & Delta-v (physics). The author has an hindex of 3, co-authored 13 publications receiving 42 citations.

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Abstract: We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = H(u)\mu &{}\quad \text {in}\ \Omega ,\\ u>0 &{}\quad \text {in}\ \Omega ,\\ u=0 &{}\quad \text {on}\ \partial \Omega . \end{array}\right. } \end{aligned}$$ Here $$\Omega $$ is an open bounded subset of $${\mathbb {R}}^N$$ ( $$N\ge 2$$ ), $$\Delta _p u:= {\text {div}}(| abla u|^{p-2} abla u)$$ ( $$1

Asymptotic behavior and existence of solutions for singular elliptic equations

Abstract: We study the asymptotic behavior, as $$\gamma $$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is $$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$ where $$\Omega $$ is an open, bounded subset of $${\mathbb {R}}^{N}$$ and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f: either strictly positive on every compactly contained subset of $$\Omega $$ or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with $$\begin{aligned} -\Delta v + \frac{| abla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$

Asymptotic behavior and existence of solutions for singular elliptic equations

Abstract: We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega, $$ where $\Omega$ is an open, bounded subset of $\RN$ and $f$ is a bounded function. We deal with the existence of a limit equation under two different assumptions on $f$: either strictly positive on every compactly contained subset of $\Omega$ or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to $$ -\Delta v + \frac{| abla v|^2}{v} = f\,\text{ in }\Omega. $$

Regularizing effect for some p-Laplacian systems

Abstract: We study existence and regularity of weak solutions for the following p -Laplacian system − Δ p u + A φ θ + 1 | u | r − 2 u = f , u ∈ W 0 1 , p ( Ω ) , − Δ p φ = | u | r φ θ , φ ∈ W 0 1 , p ( Ω ) , where Ω is an open bounded subset of R N ( N ≥ 2 ) , Δ p v ≔ div ( | ∇ v | p − 2 ∇ v ) is the p -Laplacian operator, for 1 p N , A > 0 , r > 1 , 0 ≤ θ p − 1 and f belongs to a suitable Lebesgue space. In particular, we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions.

Shape Programming of a Magnetic Elastica

Abstract: We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve ...

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

Abstract: In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0

The Dirichlet problem for singular elliptic equations with general nonlinearities

Abstract: In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle - \Delta _{1} u = h(u)f &{} \text {in}\, \Omega , \\ u\ge 0&{} \text {in}\ \Omega ,\\ u=0 &{} \text {on}\ \partial \Omega \,. \end{array}\right. } \end{aligned}$$ Here $$\Delta _{1} $$ is the 1-Laplace operator, $$\Omega $$ is a bounded open subset of $$\mathbb {R}^N$$ with Lipschitz boundary, h(s) is a continuous function which may become singular at $$s=0^{+}$$ , and f is a nonnegative datum in $$L^{N,\infty }(\Omega )$$ with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and $$f>0$$ . As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with $$p>1$$ ) as principal part is also established. The main assumptions are further discussed in order to show their optimality.