Related papers: A deterministic optimal design problem for the hea…
We consider the homogeneous heat equation in a domain $\Omega$ in $\mathbb{R}^n$ with vanishing initial data and the Dirichlet boundary condition. We are looking for solutions in $W^{r,s}_{p,q}(\Omega\times(0,T))$, where $r < 2$, $s < 1$,…
In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate…
This paper studies the state observation problems for the semilinear heat equation in R^n. We derive observation estimates for the equation using the logarithmic convexity property of the frequency function (see [12]). As an application, we…
We investigate a shape optimization problem for a heat-conducting fluid governed by a Boussinesq system. The main goal is to determine an optimal domain shape that yields a temperature distribution as uniform as possible. Initially, we…
We investigate the nonlocal energy corresponding to the $p$-oscillation of the unit normal vector for hypersurfaces, or the unit tangent vector for curves. The energy satisfies geometric inequalities with optimal constants $c(n,p)$ and…
In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $\Omega \subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in…
In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)},$$ where $1\le q<p\le+\infty$ and $\lambda_p(\Omega)$ (respectively $\lambda_q(\Omega)$) is…
We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation…
Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $\omega_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit $\varepsilon\rightarrow 0$,…
This paper concerns the final value problem for the heat equation under the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that…
The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of…
In this paper we consider a shape optimization problem in which the data in the cost functional and in the state equation may change sign, and so no monotonicity assumption is satisfied. Nevertheless, we are able to prove that an optimal…
We study thermal insulating of a bounded body $\Omega\subset \mathbb{R}^n$. Under a prescribed heat source $f\geq 0$, we consider a model of heat transfer between $\Omega$ and the environment determined by convection; this corresponds,…
We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness…
Optimal packing of spheres in $\mathbb R^d$ is studied by optimization of the energy $E$ (effective conductivity) of composites with ideally conducting spherical inclusions. It is demonstrated that the minimum of $E$ over locations of…
We consider the linear heat equation on a bounded domain. We study estimates of the derivatives, up to the second order, of the solution locally in time in the Lebesgue spaces. We give a self-contained proof of the estimates in the…
We investigate the Dirichlet boundary control of the Laplace equation, considering the control in $H^{1/2}(\partial \Omega)$, which is the natural space for Dirichlet data when the state belongs to $H^1(\Omega)$. The cost of the control is…
We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…
Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an…
Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…