Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.11011

Approximation in an optimal design problem governed by the heat equation

Kei Matsushima·Tomoyuki Oka
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper studies a two-material optimal design problem for the time-averaged duality pairing between a (possibly time-dependent) heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In general, such optimal designs are not guaranteed to exist, and geometric constraints such as the perimeter are required. As an approximation of the problem with an additional perimeter constraint, a material representation based on a level set function, together with a perturbation of the Dirichlet energy, is employed. It is then shown that optimal level set functions exist for the perturbation problem, and the corresponding minimum value converges to that of the elliptic case, thereby elucidating the long-time behavior. Furthermore, two-material domains satisfying this property are also constructed via the nonlinear diffusion-based level set method. In particular, the asymptotic behavior with respect to the perturbation parameter is clarified, and the validity of the approximation is established.

Explore related subjects

math.APmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kei Matsushima, Tomoyuki Oka. 2025-09-13. Approximation in an optimal design problem governed by the heat equation. https://arxiv.org/abs/2509.11011

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Large friction limit of compressible Navier--Stokes equations with Navier boundary conditions in a half-space

We study the large-friction limit for the three-dimensional barotropic compressible Navier-Stokes equations in a half-space. The velocity satisfies a Navier boundary condition with friction coefficient $α>0$, while the limiting problem satisfies the no-slip boundary condition. We establish estimates for local-in-time smooth solutions that are uniform in $α$ and prove strong convergence of the density and velocity as $α\to\infty$. For weak solutions, we use the Lagrangian flow maps associated with the two velocities to compare the densities and construct suitable transported test functions. This yields weak convergence to the no-slip solution. Our results provide a compressible counterpart of the large-friction limit for incompressible flows.

math.AP↗
arXiv2026

Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\LL_{\A}$ in ${\mathbb R^2}$, which involve the {physical} Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\not= 2$, the Bochner-Riesz operator $S_λ^δ(\LL_{\A})$ of order $δ$ is bounded on $L^p(\R^2)$ if and only if $δ>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient {in} the proof is to obtain the localized $L^4(\R^2)$ estimate of $S_λ^δ(\LL_{\A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_λ^δ(Δ)$ for the Laplacian $Δ$ in ${\mathbb R}^2$.

math.AP↗
arXiv2026

Solitons, scattering and blow-up for the nonlinear Schrödinger equation with combined power-type nonlinearities on $\mathbb{R}^d\times\mathbb{T}$

We investigate the long time dynamics of the nonlinear Schrödinger equation (NLS) with combined powers on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$. Different from the previously studied NLS-models with single power on the waveguide manifolds, where the non-scale-invariance is mainly due to the mixed nature of the underlying domain, the non-scale-invariance of the present model is both geometrical and structural. By considering different combinations of the nonlinearities, we establish both qualitative and quantitative properties of the soliton, scattering and blow-up solutions. As one of the main novelties of the paper compared to the previous results for the NLS with single power, we particularly construct two different rescaled families of variational problems, which leads to an NLS with single power in different limiting profiles respectively, to establish the periodic dependence results.

math.AP↗