Dirichlet-type Conditions Research Articles

SOME RECENT RESULTS ON NONLINEAR PARABOLIC PROBLEMS RELATED TO GREINER OPERATOR

Abstract. Greiner operator arise a diverse area of mathematics involving boundary value problems, harmonic analysis, quantum mechanics and electromagnetic fields. In this study, we shall investigate the absence of positive solutions for a p-Laplacian parabolic problems with a forcing term and critical potentials including Dirichlet-type conditions in a relevant Carnot Carathéodory metric ball Ω in 𝑅 2𝑛+1 with smooth boundary. Keywords: Critical exponents, Greiner operator, positive solutions.

Multiple solutions for the fractional p-Laplacian with jumping reactions

We study a nonlinear elliptic equation driven by the degenerate fractional p-Laplacian, with Dirichlet-type condition and a jumping reaction, i.e., $$(p-1)$$ -linear both at infinity and at zero but with different slopes crossing the principal eigenvalue. Under two different sets of hypotheses, entailing different types of asymmetry, we prove the existence of at least two nontrivial solutions. Our method is based on degree theory for monotone operators and nonlinear fractional spectral theory.

A Priori Estimate of Solutions of One Boundary-Value Problem in a Strip for a Higher-Order Degenerate Elliptic Equation

In this paper, we prove coercive a priori estimates of solutions of a Dirichlet-type boundary-value problem in a strip for a certain higher-order degenerate elliptic equation containing weighted derivatives of a special form up to the order 2m and ordinary partial derivatives up to the order 2k−1 under the condition 2m > 2k −1. At the boundary of the strip, Dirichlet-type conditions are imposed. A coercive a priori estimate for solutions of the problem considered in special weighted Sobolev-type spaces is obtained.

Nonexistence criteria for systems of parabolic inequalities in an annulus

We are concerned with nonexistence results for a class of systems of parabolic inequalities in (0,∞)×A, where A={x∈RN:1<|x|≤2}. The considered systems involve a singular potential function V(x)=(|x|−1)−ρ, ρ>0, in front of the power nonlinearities. Two types of inhomogeneous boundary conditions on ∂B2={x∈RN:|x|=2} are discussed: Neumann type conditions and Dirichlet type conditions. Using a unified approach, an optimal criterium of nonexistence is obtained for both cases. Our study yields naturally optimal nonexistence results for the corresponding stationary systems.

Nonlinear differential equations with perturbed Dirichlet integral boundary conditions

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green’s function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones.

Estimation Technique for a Contact Point Between two Materials in a Stationary Heat Transfer Problem

An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, of negligible diameter, made up of two consecutive sections of different, isotropic and homogeneous materials. At the left boundary, a Dirichlet type condition is imposed that represents a constant temperature source while a Robin type condition that models the heat dissipation by convection is considered at the right one. Many articles in the literature focus on thermal and stress analysis at the interface but no one is dedicated to the estimation of the contact point location between the two materials. In this work, it is assumed that the interface position is unknown. A technique to determine it from a unique noisy flow measurement at the right boundary is introduced. Necessary and sufficient conditions are derived in order to obtain the estimation of the interface point from a heat flux measured at the right boundary. Numerical solutions are obtained together with an expression for the estimation error. Moreover, an elasticity analysis is included to study the influence of data errors. The results show that our approach is useful for determining the location of the materials interface.

Fine boundary regularity for the degenerate fractional p-Laplacian

We consider a nonlocal equation driven by the fractional p-Laplacian (−Δ)ps with s∈]0,1[ and p⩾2 (degenerate case), with a bounded reaction f and Dirichlet type conditions in a smooth domain Ω. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution u of such equation exhibits a weighted Hölder regularity up to the boundary, that is, u/dΩs∈Cα(Ω‾) for some α∈]0,1[, dΩ being the distance from the boundary.

Extremal constant sign solutions and nodal solutions for the fractional p-Laplacian

We study a pseudo-differential equation driven by the degenerate fractional p-Laplacian, under Dirichlet type conditions in a smooth domain. First we show that the solution set within the order interval given by a sub-supersolution pair is nonempty, directed, and compact, hence endowed with extremal elements. Then, we prove existence of a smallest positive, a biggest negative and a nodal solution, combining variational methods with truncation techniques.

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.

H-convergence and homogenization of non-local elliptic operators in both perforated and non-perforated domains

In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem $$\mathcal{L}_\varepsilon^s u_\varepsilon =(-\nabla\cdot (A_\varepsilon(x)\nabla))^{s}u_\varepsilon=f \mbox{ in } \mathcal O, $$ with $0<s<1$, considering non-homogeneous Dirichlet type condition outside of the bounded domain $\mathcal O\subseteq \mathbb{R}^n$. We find the homogenized problem by using the $H$-convergence method, as $\varepsilon\to 0$, under standard uniform ellipticity, boundedness and symmetry assumptions on coefficients $A_\varepsilon(x)$, with the homogenized coefficients as the standard $H$-limit (cf. \cite{MT1}) of the sequence $\{A_\varepsilon\}_{\varepsilon>0}$. We also prove that the commonly referred to as \textit{the strange term} in the literature (see \cite[Chapter 4]{MT}) does not appear in the homogenized problem associated with the fractional Laplace operator $(-\Delta)^s$ in a perforated domain. Both of these results have been obtained in the class of general microstructures. Consequently, we could certify that the homogenization process, as $\varepsilon\to 0$, is stable under $s\to 1^{-}$ in the non-perforated domains, but not necessarily in the case of perforated domains.

Sobolev versus Hölder minimizers for the degenerate fractional [formula omitted]-Laplacian

We consider a nonlinear pseudo-differential equation driven by the fractional p-Laplacian (−Δ)ps with s∈(0,1) and p⩾2 (degenerate case), under Dirichlet type conditions in a smooth domain Ω. We prove that local minimizers of the associated energy functional in the fractional Sobolev space W0s,p(Ω) and in the weighted Hölder space Cs0(Ω¯), respectively, coincide.

A penalization method for the simulation of weakly compressible reacting gas-particle flows with general boundary conditions

In this paper, the Brinkman penalization method is extended to the weakly compressible formulation of the Navier–Stokes equations to study gas-particle flows with reactions and coupled heat and mass transfer. The weakly compressible approximation removes the acoustic modes from the solution while allowing a variable density. The Brinkman volume penalization method describes the solid phase as a porous medium with a vanishing permeability. The method is validated with literature benchmarks for a fixed or moving particle. Various reaction scenarios are then investigated. First, the capability of the method to deal with intra-particle diffusion and reaction is evaluated. The impact on the conversion of the particle Reynolds number based on the slip velocity is assessed, for a range of Reynolds numbers encountered in fluidization. Next, surface reactions are focused on, with infinite or finite rate. A Dirichlet-type condition is imposed on the particle surface to treat an infinite reaction rate and a high solid thermal conductivity while a finite-rate surface reaction requires Neumann and Robin surface conditions for the temperature and species mass fractions. The impact of density changes induced by heat release and molecular weight changes is assessed in these different cases. It is shown that the combination of the weakly compressible approximation and the penalization method allows to treat the presence of the solid phase and reactions in an efficient manner and to take into account the effects of compressibility in commonly encountered situations in reaction engineering.

Exact harmonic solutions to Guyer–Krumhansl-type equation and application to heat transport in thin films

A system of hyperbolic-type inhomogeneous differential equations (DE) is considered for non-Fourier heat transfer in thin films. Exact harmonic solutions to Guyer–Krumhansl-type heat equation and to the system of inhomogeneous DE are obtained in Cauchy- and Dirichlet-type conditions. The contribution of the ballistic-type heat transport, of the Cattaneo heat waves and of the Fourier heat diffusion is discussed and compared with each other in various conditions. The application of the study to the ballistic heat transport in thin films is performed. Rapid evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow evolution of its diffusive counterpart. The effect of the ballistic quasi-temperature component on the evolution of the complete quasi-temperature is explored. In this context, the influence of the Knudsen number and of Cauchy- and Dirichlet-type conditions on the evolution of the temperature distribution is explored. The comparative analysis of the obtained solutions is performed.

A Heat–Viscoelastic Structure Interaction Model with Neumann and Dirichlet Boundary Control at the Interface: Optimal Regularity, Control Theoretic Implications

We consider a heat---structure interaction model where the structure is subject to viscoelastic (strong) damping, and where a (boundary) control acts at the interface between the two media in Neumann-type or Dirichlet-type conditions. For the boundary (interface) homogeneous case, the free dynamics generates a s.c. contraction semigroup which, moreover, is analytic on the energy space and exponentially stable here Lasiecka et al. (Pure Appl Anal, to appear). If $${\mathcal {A}}$$A is the free dynamics operator, and $${\mathcal {B}}_N$$BN is the (unbounded) control operator in the case of Neumann control acting at the interface, it is shown that $${\mathcal {A}}^{-\frac{1}{2}}{\mathcal {B}}_N$$A-12BN is a bounded operator from the interface measured in the $$\mathbf{L}^2$$L2-norm to the energy space. We reduce this problem to finding a characterization, or at least a suitable subspace, of the domain of the square root $$(-{\mathcal {A}})^{1/2}$$(-A)1/2, i.e., $${\mathcal {D}}((-{\mathcal {A}})^{1/2})$$D((-A)1/2), where $${\mathcal {A}}$$A has highly coupled boundary conditions at the interface. To this end, here we prove that $${\mathcal {D}}((-{\mathcal {A}})^{\frac{1}{2}})\equiv {\mathcal {D}}((-{\mathcal {A}}^*)^{\frac{1}{2}})\equiv V$$D((-A)12)?D((-A?)12)?V, with the space V explicitly characterized also in terms of the surviving boundary conditions. Thus, this physical model provides an example of a matrix-valued operator with highly coupled boundary conditions at the interface, where the so called Kato problem has a positive answer. (The well-known sufficient condition Lions in J Math Soc JAPAN 14(2):233---241, 1962, Theorem 6.1 is not applicable.) After this result, some critical consequences follow for the model under study. We explicitly note here three of them: (i) optimal (parabolic-type) boundary $$\rightarrow $$? interior regularity with boundary control at the interface; (ii) optimal boundary control theory for the corresponding quadratic cost problem; (iii) min---max game theory problem with control/disturbance acting at the interface. On the other hand, if $${\mathcal {B}}_D$$BD is the (unbounded) control operator in the case of Dirichlet control acting at the interface, it is then shown here that $${\mathcal {A}}^{-1}{\mathcal {B}}_D$$A-1BD is a bounded operator from the interface measured this time in the $$\mathbf{H}^{\frac{1}{2}}$$H12-norm to the energy space. Similar consequences follow.

A Dirichlet problem on the half-line for nonlinear equations with indefinite weight

We study the existence of positive solutions on the half-line \([0,\infty )\) for the nonlinear second-order differential equation $$\begin{aligned} \bigl (a(t)x^{\prime }\bigr )^{\prime }+b(t)F(x)=0,\quad t \ge 0, \end{aligned}$$ satisfying Dirichlet-type conditions, say \(x(0)=0\), \(\lim _{t\rightarrow \infty }x(t)=0\). The function b is allowed to change sign, and the nonlinearity F is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which b is a periodic function for large t or it is unbounded from below.

A generalized Poisson solver for first-principles device simulations

Electronic structure calculations of atomistic systems based on density functional theory involve solving the Poisson equation. In this paper, we present a plane-wave based algorithm for solving the generalized Poisson equation subject to periodic or homogeneous Neumann conditions on the boundaries of the simulation cell and Dirichlet type conditions imposed at arbitrary subdomains. In this way, source, drain, and gate voltages can be imposed across atomistic models of electronic devices. Dirichlet conditions are enforced as constraints in a variational framework giving rise to a saddle point problem. The resulting system of equations is then solved using a stationary iterative method in which the generalized Poisson operator is preconditioned with the standard Laplace operator. The solver can make use of any sufficiently smooth function modelling the dielectric constant, including density dependent dielectric continuum models. For all the boundary conditions, consistent derivatives are available and molecular dynamics simulations can be performed. The convergence behaviour of the scheme is investigated and its capabilities are demonstrated.

Scalar boundary conditions in hyperscaling violating geometry

We study the possible boundary conditions of scalar field modes in a hyperscaling violation (HV) geometry with Lifshitz dynamical exponent z(z≥1) and hyperscaling violation exponent θ (θ≠0). For the case with θ>0, we show that in the parameter range 1≤z≤2, −z+d−1<θ≤(d−1)(z−1) or z>2, −z+d−1<θ≤d−1, the boundary conditions have different types, including the Neumann, Dirichlet and Robin conditions, while in the range θ≤−z+d−1, only Dirichlet type condition can be set. In particular, we further confirm that the mass of the scalar field does not play any role in determining the possible boundary conditions for θ>0, which has been addressed in Ref. [1]. Meanwhile, we also carry out the parallel investigation in the case with θ<0. We find that for m2<0, three types of boundary conditions are available, but for m2>0, only one type is available.

Problem with two-point conditions for parabolic equation of second order on time

The correctness of the problem with two-point conditions on time variable and Dirichlet-type conditions on spetial coordinates for the linear parabolic equations are established. The metric theorem about estimate from below of small denominators of the problem is proved.

A new analytical solution solved by triple series equations method for constant-head tests in confined aquifers

The constant-head pumping tests are usually employed to determine the aquifer parameters and they can be performed in fully or partially penetrating wells. Generally, the Dirichlet condition is prescribed along the well screen and the Neumann type no-flow condition is specified over the unscreened part of the test well. The mathematical model describing the aquifer response to a constant-head test performed in a fully penetrating well can be easily solved by the conventional integral transform technique under the uniform Dirichlet-type condition along the rim of wellbore. However, the boundary condition for a test well with partial penetration should be considered as a mixed-type condition. This mixed boundary value problem in a confined aquifer system of infinite radial extent and finite vertical extent is solved by the Laplace and finite Fourier transforms in conjunction with the triple series equations method. This approach provides analytical results for the drawdown in a partially penetrating well for arbitrary location of the well screen in a finite thickness aquifer. The semi-analytical solutions are particularly useful for the practical applications from the computational point of view.

Amplitude equations for a linear wave equation in a weakly curved pipe

We study boundary effects in a linear wave equation with Dirichlet-type conditions in a weakly curved pipe. The coordinates in our pipe are prescribed by a given small curvature with finite range, with the pipe's cross section being circular. Based on the straight pipe case, a perturbative analysis by which the boundary value conditions are exactly satisfied is employed. As such an analysis, we decompose the wave equation into a set of ordinary differential equations perturbatively. We show the conditions when secular terms due to the curved boundary appear in the naive peturbative analysis. In eliminating such a secularity with a singular perturbation method, we derive amplitude equations and show that the eigenfrequencies in time are shifted due to the curved boundary.