0000000000077441
AUTHOR
Barbara Brandolini
showing 31 related works from this author
Serrin-Type Overdetermined Problems: an Alternative Proof
2008
We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely the Hessian equations. In the case of the Poisson equation, our proof is alternative to the proofs proposed by Serrin (moving planes) and by Weinberger. Moreover, our proof makes no direct use of the maximum principle while it sheds light on a relation between the Serrin problem and the isoperimetric inequality.
Sharp estimates for eigenfunctions of a Neumann problem
2009
In this paper we provide some bounds for the eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of R^n. To this aim we use the so-called symmetrization techniques and the obtained estimates are asymptotically sharp, at least in the bidimensional case, when the isoperimetric constant relative to Ω goes to 0.
Shape optimization for monge-ampére equations via domain derivative
2011
In this note we prove that, if $\Omega$ is a smooth, strictly convex, open set in $R^n$ $(n \ge 2)$ with given measure, the $L^1$ norm of the convex solution to the Dirichlet problem $\det D^2 u=1$ in $\Omega$, $u=0$ on $\partial\Omega$, is minimum whenever $\Omega$ is an ellipsoid.
A sharp estimate of the extinction time for the mean curvature flow
2007
We establish a pointwise comparison result for a nonlinear degenerate elliptic Dirichlet problem using an isoperimetric inequality involving the total mean curvature. In particular this result provides a sharp estimate for the extinction time of a class of compact surfaces, wider than the convex one, moving by mean curvature flow. Finally we present numerical experiments to compare our estimate with those known in literature.
Symmetrization for singular semilinear elliptic equations
2012
In this paper, we prove some comparison results for the solution to a Dirichlet problem associated with a singular elliptic equation and we study how the summability of such a solution varies depending on the summability of the datum f. © 2012 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.
Characterization of ellipsoids through an overdetermined boundary value problem of Monge–Ampère type
2014
Abstract The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non-standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.
Viscosity solutions of the Monge-Ampère equation with the right hand side in Lp
2007
We compare various notions of solutions of Monge-Ampère equations with discontinuous functions on the right hand side. Precisely, we show that the weak solutions defined by Trudinger can be obtained by the vanishing viscosity approximation method. Moreover, we investigate existence and uniqueness of Lp-viscosity solutions.
Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems
2013
In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.
Some applications of the Chambers isoperimetric inequality
2022
In this paper, using the Chambers isoperimetric inequality, we introduce the notion of weighted rearrangement of a function associated to the measure $f dx$, where $f(x)=e^{g(|x|)}$ for $x \in \mathbb{R}^n}$, with $g$ smooth, convex and even. Then we give some of its applications to variational inequalities and PDEs via weighted symmetrization.
An upper bound for nonlinear eigenvalues on convex domains by means of the isoperimetric deficit
2010
We prove an upper bound for the first Dirichlet eigenvalue of the p-Laplacian operator on convex domains. The result implies a sharp inequality where, for any convex set, the Faber-Krahn deficit is dominated by the isoperimetric deficit.
Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions
2019
Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ \Omega_2 \subset \Omega_1 \subset \mathbb{R}^n$: $$ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Ome…
The equality case in a Poincaré–Wirtinger type inequality
2016
It is known that, for any convex planar set W, the first non-trivial Neumann eigenvalue μ1 (Ω) of the Hermite operator is greater than or equal to 1. Under the additional assumption that Ω is contained in a strip, we show that β1 (Ω) = 1 if and only if Ω is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
Estimates for Sums of Eigenvalues of the Free Plate via the Fourier Transform
2017
Using the Fourier transform, we obtain upper bounds for sums of eigenvalues of the free plate.
A symmetrization result for Monge–Ampère type equations
2007
In this paper we prove some comparison results for Monge–Ampere type equations in dimension two. We also consider the case of eigenfunctions and we derive a kind of “reverse” inequalities. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Sharp Poincaré inequalities in a class of non-convex sets
2018
Let $gamma$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $gamma$, within a suitable distance $delta$ of $gamma$. Denote by $mu_1^{odd}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $gamma$ satisfies some simple geometric conditions, then $mu_1^{odd}(D)$ can be sharply estimated from below in terms of the length of $gamma$ , its curvature, and $delta$. Moreover, we give explicit conditions on $delta$ that ensure $mu_1^{odd}(D)=mu_1(D)$. Finally, we can extend our bound on $mu_1^{odd}(D)$ to a …
An optimal Poincaré-Wirtinger inequality in Gauss space
2013
International audience; Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$% -dimensional Gaussian measure.
A remark on the radial minimizer of the Ginzburg-Landau functional
2014
Let Omega subset of R-2 be a bounded domain with the same area as the unit disk B-1 and letE-epsilon(u, Omega) = 1/2 integral(Omega) vertical bar del u vertical bar(2) dx + 1/4 epsilon(2) integral(Omega) (vertical bar u vertical bar(2) - 1)(2) dxbe the Ginzburg-Landau functional. Denote by (u) over tilde (epsilon) the radial solution to the Euler equation associated to the problem min {E-epsilon (u, B-1) : u vertical bar(partial derivative B1) = x} and byK = {v = (v(1), v(2)) is an element of H-1 (Omega; R-2) : integral(Omega) v(1) dx = integral(Omega) v(2) dx = 0,integral(Omega) vertical bar v vertical bar(2) dx >= integral(B1) vertical bar(u) over tilde vertical bar(2) dx}.In this note…
Symmetry breaking in a constrained cheeger type isoperimetric inequality
2015
We study the optimal constant in a Sobolev inequality for BV functions with zero mean value and vanishing outside a bounded open set. We are interested in finding the best possible embedding constant in terms of the measure of the domain alone. We set up an optimal shape problem and we completely characterize the behavior of optimal domains.
On the stability of the Serrin problem
2008
We investigate stability issues concerning the radial symmetry of solutions to Serrin's overdetermined problems. In particular, we show that, if $u$ is a solution to $\Delta u=n$ in a smooth domain $\Omega \subset \rn$, $u=0$ on $\partial\Omega$ and $|Du|$ is close to 1 on $\partial\Omega$, then $\Omega$ is close to the union of a certain number of disjoint unitary balls.
Stability of radial symmetry for a Monge-Ampère overdetermined problem
2008
Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data. © 2008 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.
Local behaviour of singular solutions for nonlinear elliptic equations in divergence form
2012
We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and $${\mathcal{A}}$$ is a positive C 1(0,1] function which is regularly varying at zero with index $${\vartheta}$$ in (2−N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if $${\Phi\not\in L^q(B_1(0))}$$ , where $${\Phi}$$ denotes the fundamental solution of $${-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}$$ in $${\mathcal D'(B_1(0))}$$ and δ0 is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of al…
New isoperimetric estimates for solutions to Monge - Ampère equations
2009
Abstract We prove some sharp estimates for solutions to Dirichlet problems relative to Monge–Ampere equations. Among them we show that the eigenvalue of the Dirichlet problem, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result falls in the class of affine isoperimetric inequalities and shows that the eigenvalue of the Monge–Ampere operator behaves just the contrary of the first eigenvalue of the Laplace operator.
Perimeter symmetrization of some dynamic and stationary equations involving the Monge-Ampère operator
2017
We apply the perimeter symmetrization to a two-dimensional pseudo-parabolic dynamic problem associated to the Monge-Ampere operator as well as to the second order elliptic problem which arises after an implicit time discretization of the dynamical equation. Curiously, the dynamical problem corresponds to a third order operator but becomes a singular second order parabolic equation (involving the 3-Laplacian operator) in the class of radially symmetric convex functions. Using symmetrization techniques some quantitative comparison estimates and several qualitative properties of solutions are given.
On the Symmetry of Solutions to a k-Hessian Type Equation
2013
Abstract In this note we prove that if u is a negative solution to a nonlinear elliptic equation involving a Hessian operator, and u is zero on the boundary of a ball, then u is radially symmetric and increasing along the radii.
Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift
2015
Abstract This paper deals with the eigenvalue problem for the operator L = − Δ − x ⋅ ∇ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue λ k of L under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any c > 0 and k ∈ N the following minimization problem min { λ k ( Ω ) : Ω quasi-open set , ∫ Ω e | x | 2 / 2 d x ≤ c } has a solution.
Existence and comparison results for a singular semilinear elliptic equation with a lower order term
2014
This paper deals with the homogeneous Dirichlet problem for a singular semilinear elliptic equation with a first order term. When the datum is bounded we prove an existence result and we show that any solution can be compared with the solution to a suitable symmetrized problem.
A sharp estimate for Neumann eigenvalues of the Laplace-Beltrami operator for domains in a hemisphere
2018
Here, we prove an isoperimetric inequality for the harmonic mean of the first [Formula: see text] non-trivial Neumann eigenvalues of the Laplace–Beltrami operator for domains contained in a hemisphere of [Formula: see text].
Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem
2011
Abstract We determine the shape which minimizes, among domains with given measure, the first eigenvalue of a nonlocal operator consisting of a perturbation of the standard Dirichlet Laplacian by an integral of the unknown function. We show that this problem displays a saturation behaviour in that the corresponding value of the minimal eigenvalue increases with the weight affecting the average up to a (finite) critical value of this weight, and then remains constant. This critical point corresponds to a transition between optimal shapes, from one ball as in the Faber–Krahn inequality to two equal balls.
Comparison results for Hessian equations via symmetrization
2007
where the λ’s are the eigenvalues of the Hessian matrix D2u of u and Sk is the kth elementary symmetric function. For example, for k = 1, S1(Du) = 1u, while, for k = n, Sn(D 2u) = detD2u. Equations involving these operators, and some more general equations of the form F(λ1, . . . , λn) = f in , (1.2) have been widely studied by many authors, who restrict their considerations to convenient cones of solutions with respect to which the operator in (1.2) is elliptic. Following [25] we define the cone 0k of ellipticity for (1.1) to be the connected component containing the positive cone 0 = {λ ∈ R : λi > 0 ∀i = 1, . . . , n} of the set where Sk is positive. Thus 0k is an open, convex, symmetric…
Anisotropic elliptic equations with gradient-dependent lower order terms and L^1 data
2023
<abstract><p>We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $ \mathcal A $, the prototype of which is $ \mathcal A u = -\sum_{j = 1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u) $ with $ p_j &gt; 1 $ for all $ 1\leq j\leq N $ and $ \sum_{j = 1}^N (1/p_j) &gt; 1 $. As a novelty in this paper, our lower order terms involve a new class of operators $ \mathfrak B $ such…
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