arXiv subjects
Publications and source records attributed to Stefano Biagi.
We study mixed local and nonlocal elliptic equation with a variable coefficient $ρ$. Under suitable assumptions on the behaviour at infinity of $ρ$, we obtain uniqueness of solutions belonging to certain weighted Lebsgue spaces, with a weight depending on the coefficient $ρ$. The hypothesis on $ρ$ is optimal; indeed, when it fails we get nonuniqueness of solutions. We also investigate the parabolic counterpart of such equation.
We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is $u\equiv 0$. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.
We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such that the corresponding model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group operation in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients $a_{ij}$ are bounded and Hölder continuous in space (w.r.t. some distance induced by $\mathcal{L}$ in $\mathbb{R}^{N}$) and only bounded measurable in time; the matrix $\{ a_{ij}\}_{i,j=1}^{q}$ is symmetric and uniformly positive on $\mathbb{R}^{q}$. We prove "partial Schauder a priori estimates" the kind $$ \sum_{i,j=1}^{q}\Vert\partial_{x_{i}x_{j}}^{2}u\Vert_{C_{x}^α(S_{T})}+\Vert Yu\Vert_{C_{x}^α(S_{T})}\leq c\left\{ \Vert\mathcal{L}u\Vert _{C_{x}^α(S_{T})}+\Vert u\Vert_{C^{0}(S_{T})}\right\} $$ for suitable functions $u$, where $$ \Vert f\Vert_{C_{x}^α(S_{T})}=\sup_{t\leq T}\sup_{x_{1},x_{2}\in\mathbb{R}^{N},x_{1}\neq x_{2}}\frac{\left\vert f\left( x_{1},t\right) -f\left( x_{2},t\right) \right\vert }{\left\Vert x_{1}-x_{2}\right\Vert ^α}. $$ We also prove that the derivatives $\partial_{x_{i}x_{j}}^{2}u$ are locally Hölder continuous in space and time while $\partial_{x_{i}}u$ and $u$ are globally Hölder continuous in space and time.
We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.
In this paper we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e., $$\mathcal{L}_{p,s} = -Δ_p + (-Δ)^s_p.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest.
We continue the study of positive singular solutions of PDEs arising from double phase functionals started in [6]. In particular, we consider the case $p<q < 2$, and we relax the assumption on the capacity of the singular set using an intrinsic notion of capacity.
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by $p$-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.
We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.
Given a bounded open set $Ω\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $Ω$. We prove that the second eigenvalue $λ_2(Ω)$ is always strictly larger than the first eigenvalue $λ_1(B)$ of a ball $B$ with volume half of that of $Ω$. This bound is proven to be sharp, by comparing to the limit case in which $Ω$ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.
Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t}% $$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded Hölder continuous functions on $\mathbb{R}^{1+n}$, with respect to the "parabolic" distance induced by the vector fields. We prove the existence of a global heat kernel $Γ(\cdot;s,y)\in C_{X,\mathrm{loc}}^{2,α}(\mathbb{R}^{1+n}\setminus\{(s,y)\})$ for $\mathcal{H}$, such that $Γ$ satisfies two-sided Gaussian bounds and $\partial_{t}Γ, X_{i}Γ,X_{i}X_{j}Γ$ satisfy upper Gaussian bounds on every strip $[0,T]\times\mathbb{R}^n$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H}$, and a standard Harnack inequality for the corresponding stationary operator $$ \mathcal{L}:=\sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j}. $$ with Hölder continuos coefficients.
We study the existence of non-zero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with non-local functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
We prove Pohozaev-type identities for smooth solutions of Euler-Lagrange equations of second and fourth order that arise from functional depending on homogeneous Hörmander vector fields. We then exploit such integral identities to prove non-existence results for the associated boundary value problems.
In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form $-Δ+(-Δ)^s$, with $s\in(0,1)$. We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.
Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth Hörmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$, while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel $Γ(t,x;s,y)$ of $\mathcal{H}$, in terms of the Carnot-Carathéodory distance induced by $X$ on $\mathbb{R}^{n}$, as well as global upper Gaussian estimates for the $t$- or $X$-derivatives of any order of $Γ$. From the Gaussian bounds we derive the unique solvability of the Cauchy problem for a possibly unbounded continuous initial datum satisfying exponential growth at infinity. Also, we study the solvability of the H-Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of $\mathcal{H}u=0$.
Let $\mathcal{L}=\sum_{j=1}^{m}X_{j}^{2}$ be a Hörmander sum of squares of vector fields in $\mathbb{R}^{n}$, where any $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$. Then $\mathcal{L}$ is known to admit a global fundamental solution $Γ(x;y)$, that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $\mathbb{R}^{n}\times \mathbb{R}^{p}$, equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of $Γ$, in terms of the Carnot-Carathéodory distance induced by $X=\{X_{1},\ldots ,X_{m}\}$ on $\mathbb{R}^{n}$, as well as global pointwise (upper) estimates for the $X$-derivatives of any order of $Γ$, together with suitable integral representations of these derivatives. The least dimensional case $n=2$ presents several peculiarities which are also investigated. Applications to the potential theory for $\mathcal{L}$ and to singular-integral estimates for the kernel $X_{i}X_{j}Γ$ are also provided. Finally, most of the results about $Γ$ are extended to the case of Hörmander operators with drift $\sum_{j=1}^{m}X_{j}^{2}+X_{0}$, where $X_{0}$ is $2$-homogeneous and $X_{1},...,X_{m}$ are $1$-homogeneous.
We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type $$\big(Φ(k(t)\,x'(t))\big)' + f(t,\mathcal{G}_x(t))\,ρ(t, x'(t)) = 0$$ on a compact interval $[a,b]$. These equations are quite general due to the presence of a strictly increasing homeomorphism $Φ$, the so-called $Φ$-Laplacian operator, of a nonnegative function $k$, which may vanish on a set of null measure, and moreover of a functional term $\mathcal{G}_x$. We look for solutions, in a suitable weak sense, which belong to the Sobolev space $W^{1,1}([a,b])$. Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.
We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of non-negative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.