Bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws

kth Order convergence for semilinear elliptic boundary value problems

Uniqueness for sublinear boundary value problems

The concept of hypoellipticity for degenerate elliptic boundary value problems is defined, and its relation with the hypoellipticity of certain pseudo-differential operators on the boundary is discussed (for second order equations). A theorem covering smoothness of solutions of boundary value problems such as a ( x ) ∂ u / ∂ n + b ( x ) u = f ( x ) a(x)\partial u/\partial n + b(x)u = f(x) for the Laplace equation is proved. An almost complete characterization of hypoelliptic boundary value problems for elliptic second order equations in two dimensions is given via analysis of hypoelliptic pseudo-differential operators in one variable.

The asymptotic expansion of the heat kernel $$ \Theta {\left( t \right)} = {\sum\limits_{j = 1}^\infty {\exp {\left( { - t\lambda _{j} } \right)}} } $$ where $$ {\left\{ {\lambda _{j} } \right\}}^{\infty }_{{j = 1}} $$ are the eigenvalues of the negative Laplacian $$ - \Delta _{n} = - {\sum\limits_{k = 1}^n {{\left( {\frac{\partial } {{\partial x^{k} }}} \right)}} }^{2} $$ in R n (n = 2 or 3) is studied for short-time t for a general bounded domain Ω with a smooth boundary $$ \partial \Omega $$ . In this paper, we consider the case of a finite number of the Dirichlet conditions φ = 0 on Γ i (i = 1, ..., J) and the Neumann conditions $$ \frac{{\partial \phi }} {{\partial \upsilon _{i} }} = 0 $$ on Γ i (i = J +1, · · · , k) and the Robin conditions $$ {\left( {\frac{\partial } {{\partial \upsilon _{i} }} + \gamma _{i} } \right)}\phi = 0 $$ on Γ i (i = k+1, · · ·,m) where γ i are piecewise smooth positive impedance functions, such that $$ \partial \Omega $$ consists of a finite number of piecewise smooth components Γ i (i = 1, · · ·,m) where $$ \partial \Omega = {\bigcup\limits_{i = 1}^m {\Gamma _{i} } } $$ . We construct the required asymptotics in the form of a power series over t. The senior coefficients in this series are specified as functionals of the geometric shape of the domain Ω. This result is applied to calculate the one-particle partition function of a “special ideal gas”, i. e., the set of non-interacting particles set up in a box with Dirichlet, Neumann and Robin boundary conditions for the appropriate wave function. Calculation of the thermodynamic quantities for the ideal gas such as the internal energy, pressure and specific heat reveals that these quantities alone are incapable of distinguishing between two different shapes of the domain. This conclusion seems to be intuitively clear because it is based on a limited information given by a one-particle partition function; nevertheless, its formal theoretical motivation is of some interest.

Multiple solutions for resonant semilinear elliptic problems in [formula omitted

Cesari [7] & I in a preceding paper established conditions sufficient that a semilinear elliptic boundary-value problem, possibly not selfadjoint shall have a solution.

The Dirichlet and Neumann conditions are commonly employed as boundary conditions for the heat equation, yet their legitimacy is debatable in certain scenarios. This paper aims to demonstrate that, in fact, diffusion laws autonomously select boundary conditions. To illustrate this, we incorporate the bounded domain into a larger domain with a diffusivity parameter ϵ > 0 and examine the solution’s behavior at the interface. Our findings reveal that homogeneous Neumann or Dirichlet boundary conditions emerge as ϵ → 0, contingent upon the type of the heterogeneous diffusion.

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

On the uniqueness of the solution for a semi-linear elliptic boundary value problem of the membrane MEMS device for reconstructing the membrane profile in absence of ghost solutions

Multiple solutions for semilinear elliptic boundary value problems with double resonance

In this paper we develop a reconstruction algorithm for the solution of an inverse boundary value problem dealing with a semilinear elliptic partial differential equation of interest in cardiac electrophysiology. The goal is the detection of small inhomogeneities located inside a domain , where the coefficients of the equation are altered, starting from observations of the solution of the equation on the boundary . Exploiting theoretical results recently achieved in [], we implement a reconstruction procedure based on the computation of the topological gradient of a suitable cost functional. Numerical results obtained for several test cases finally assess the feasibility and the accuracy of the proposed technique.

We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\varOmega \subset{\mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $\varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $\text{H}^{1}(\varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems, at last we get up to six nontrivial solutions.

When quantum field theory is formulated in terms of transition amplitudes between initial and final configurations, it is important to impose mathematically consistent boundary conditions on the fields. The choice of boundary conditions is also important when dealing with elliptic complexes and index theorems on bounded manifolds. Although Dirichlet and Neumann boundary conditions are adequate for many purposes, they are not always appropriate. However, a number of the remaining cases can be handled by imposing Dirichlet conditions on certain field components and Neumann conditions on the others. Quantum gravity and supergravity, spinor field theory, and various elliptic complexes all lend themselves to mixed boundary conditions of this sort. The mathematical significance of mixed boundary conditions is investigated, and a simple rule is obtained for determining which field components should obey which boundary conditions. It will be shown that gauge transformations on a bounded manifold give rise to intrinsic gauge transformations within the boundary. This result is used to define gauge-invariant and BRS-invariant quantum amplitudes for boundary configurations. The most general form for mixed boundary conditions on the Dirac operator and the de Rham complex will also be derived.

A thermal immersed boundary–lattice Boltzmann method for moving-boundary flows with Dirichlet and Neumann conditions