In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the L.sup.infinity and L.sup.p norms, for p greater than or equal ...to 1, of the torsion function, seen as a functional on a bounded simply connected open set OMEGA subset ??, and prove that the balls are critical shapes for these functionals, when the volume of OMEGA is preserved. KEYWORDS.--Torsion problem, Robin boundary conditions, shape derivative. 2020 MATHEMATICS SUBJECT CLASSIFICATION. - 35J05, 35J15, 35J20, 35J25.
In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions, where the velocity satisfies additional assumptions in fractional Sobolev ...spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.
In this paper we consider PDE’s problems involving the anisotropic Laplacian operator, with Robin boundary conditions. By means of Talenti techniques, widely used in the last decades, we prove a ...comparison result between the solutions of the above-mentioned problems and the solutions of the symmetrized ones. As a consequence of these results, a Bossel–Daners type inequality can be shown in dimension 2.
On a Steklov-Robin eigenvalue problem Gavitone, Nunzia; Sannipoli, Rossano
Journal of mathematical analysis and applications,
10/2023, Letnik:
526, Številka:
2
Journal Article
Recenzirano
Odprti dostop
In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider Ω=Ω0∖B‾r, where Br is the ball centered at the origin with radius r>0 and ...Ω0⊂Rn, n⩾2, is an open, bounded set with Lipschitz boundary, such that B‾r⊂Ω0. We impose a Steklov condition on the outer boundary and a Robin condition involving a positive L∞ function β(x) on the inner boundary. Then, we study the first eigenvalue σβ(Ω) and its main properties. In particular, we investigate the behavior of σβ(Ω) when we let vary the L1-norm of β and the radius of the inner ball. Furthermore, we study the asymptotic behavior of the corresponding eigenfunctions when β is a positive parameter that goes to infinity.
In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions with velocity satisfying additional assumptions in fractional Sobolev spaces ...with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.
In this paper we consider PDE's problems involving the anisotropic Laplacian operator, with Robin boundary conditions. By means of Talenti techniques, widely used in the last decades, we prove a ...comparison result between the solutions of the above-mentioned problems and the solutions of the symmetrized ones. As a consequence of these results, a Bossel-Daners type inequality can be shown in dimension 2.
In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the \(L^{\infty}\) and \(L^p\) norms, for \(p\ge 1\), of the ...torsion function, seen as a functional on a bounded simply connected open set \(\Omega \subset \mathbb{R}^n\), and prove that the balls are critical shapes for these functionals, when the volume of \(\Omega\) is preserved.
In this paper we deal with a weighted eigenvalue problem for the anisotropic \((p,q)\)-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and prove a ...reverse H\"older type inequality for the corresponding eigenfunctions.
In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider \(\Omega=\Omega_0 \setminus \overline{B}_{r}\), where \(B_{r}\) is the ball ...centered at the origin with radius \(r>0\) and \(\Omega_0\subset\mathbb{R}^n\), \(n\geq 2\), is an open, bounded set with Lipschitz boundary, such that \(\overline{B}_{r}\subset \Omega_0\). We impose a Steklov condition on the outer boundary and a Robin condition involving a positive \(L^{\infty}\)-function \(\beta(x)\) on the inner boundary. Then, we study the first eigenvalue \(\sigma_{\beta}(\Omega)\) and its main properties. In particular, we investigate the behaviour of \(\sigma_{\beta}(\Omega)\) when we let vary the \(L^1\)-norm of \(\beta\) and the radius of the inner ball. Furthermore, we study the asymptotic behaviour of the corresponding eigenfunctions when \(\beta\) is a positive parameter that goes to infinity.
In this paper we consider, by means of a precise spectral analysis, the 3D Navier-Stokes equations endowed with Navier slip-with-friction boundary conditions. We study the problem in a very simple ...geometric situation as the region between two parallel planes, with periodicity along the two planes. This setting, which is often used in the theory of boundary layers, requires some special treatment for what concerns the functional setting and allows us to characterize in a rather explicit manner eigenvalues and eigenfunctions of the associated Stokes problem. These, will be then used in order to identify infinite dimensional classes of data leading to global strong solutions for the corresponding evolution Navier-Stokes equations.
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