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On a manifoldXwith boundary and bounded geometry we consider a strongly elliptic second order operatorAtogether with a degenerate boundary operatorTof the form$$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$T=φ0γ0+φ1γ1. Here$$\gamma _0$$γ0and$$\gamma _1$$γ1denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that$$\varphi _0, \varphi _1\ge 0$$φ0,φ1≥0, and$$\varphi _0+\varphi _1\ge c$$φ0+φ1≥c, for some$$c>0$$c>0, where either$$\varphi _0,\varphi _1\in C^{\infty }_b(\partial X)$$φ0,φ1∈Cb∞(∂X)or$$\varphi _0=1 $$φ0=1and$$\varphi _1=\varphi ^2$$φ1=φ2for some$$\varphi \in C^{2+\tau }(\partial X)$$φ∈C2+τ(∂X),$$\tau >0$$τ>0. We also assume that the highest order coefficients ofAbelong to$$C^\tau (X)$$Cτ(X)and the lower order coefficients are in$$L_\infty (X)$$L∞(X). We show that the$$L_p(X)$$Lp(X)-realization ofAwith respect to the boundary operatorThas a bounded$$H^\infty $$H∞-calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.

We consider a quasilinear elliptic boundary value problem posed in a three-dimensional bounded domain containing thin vertical strips constructed on horizontal iterated von Koch curves. We study the asymptotic behavior of this problem as the width of strips tends to zero and the sequence of iterated curves converges in the Hausdorff metric to the von Koch fractal curve. We derive the effective energy of the structure with respect to a critical size of the boundary layers taking place in the neighbourhoods of the strips. This energy contains new properties implying a nonlocal term and a singular integral energy supported within the fractal curve.

In this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

We investigate the multiplicity of solutions for the one-dimensional elliptic Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems depending on the source terms wh...

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ \left \{\begin {array}{ll} \displaystyle -{\Delta }_{p} u= \frac {f}{u^{\gamma }} + g u^{q} & \text { in } {\Omega }, \\ u = 0 & \text {on } \partial {\Omega }, \end {array}\right . $$ − Δ p u = f u γ + g u q in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N , Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.

We study the homogeneous Dirichlet problem for a second-order elliptic equation with a nonlinearity discontinuous in the state variable in the resonance case. A class of resonance problems that does not overlap with the previously investigated class of strongly resonance problems is singled out. Using the variational method, we establish a theorem on the existence of at least three nontrivial solutions of the problem under study (the zero is its solution). In this case, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on the discontinuities of the nonlinearity only on a set of measure zero. We give an example of a nonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition is obtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest. Applications of the theorem to problems with a parameter are considered. The existence of nontrivial (including semiregular) solutions of the problem with a parameter for an elliptic equation with a discontinuous nonlinearity for all positive values of the parameter is established.

In this paper we will prove in Musielak–Orlicz spaces, the existence of renomalized solution for nonlinear elliptic equations of Leray-Lions type, in the case where the Musielak–Orlicz function $$ \varphi $$ doesn’t satisfy the $$\Delta _{2}$$ condition while the right hand side f belongs to $$W^{-1} E_{\psi }(\Omega )$$ .

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting , we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray , where 0$?> . As an application, we consider the Gol’dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.

The paper is concerned with estimates of the difference between a given function and the exact solution of an elliptic boundary value problem. Estimates of this type have been derived earlier in terms of the natural energy norm. In this work, an approach is proposed to obtain stronger measures of the deviation and relevant estimates applicable if the exact solution and the approximation have additional regularity (with respect to the order of integrability). These measures include the standard energy norm as a simple special case. A general approach is proposed to construct various measures based on using an auxiliary variational problem. Two classes of measures whose properties are close to those of the $${{L}^{q}}$$ and $${{L}^{\infty }}$$ norms are studied in more detail. Their properties are established, and explicitly computable two-sided estimates (minorants and majorants) that involve only known functions are constructed.

We study the properties of solutions in special weighted spaces to a nonhomogeneous boundary value problem in a planar angle for a singular elliptic equation of the second order with the differential Bessel operator ∂2/∂y2 + k∂/(y∂y), k > 0, one of the variables. Under some constraints on the weight exponents, the boundary value problem is correctly solvable. We establish a relation connecting the solutions of the problem belonging to the function spaces with different weights.

This chapter is devoted to general boundary value problems for second-order elliptic differential operators. We begin in Sect. 6.1 with a summary of the basic facts about existence, uniqueness and regularity of solutions of the Dirichlet problem in the framework of Holder spaces. In Sect 6.2, using the calculus of pseudo-differential operators, we prove existence, uniqueness and regularity theorems for the Dirichlet problem in the framework of Sobolev spaces. In Sect. 6.3 we formulate general boundary value problems, and show that these problems can be reduced to the study of pseudo-differential operators on the boundary. The virtue of this reduction is that there is no difficulty in taking adjoints after restricting the attention to the boundary, whereas boundary value problems in general do not have adjoints. This allows us to discuss the existence theory more easily.

In this paper, we consider an elliptic boundary value problem on a domain with regular boundary and discretize it with isoparametric finite elements of order $k\geq1$. We show optimal order of convergence of the isoparametric finite element solution in the $W^{1,\infty}$-norm. As an intermediate step, we derive stability and convergence estimates of optimal order $k$ for a (generalized) Ritz map.

For problems involving aseismic ground deformation in seismically active areas, numerical approaches based on the finite element scheme with discontinuity have been developed. We emphasise the utility of such numerical techniques in tackling geodynamics problems. In a viscoelastic half space representing the lithosphere-asthenosphere system, a long strike-slip fault is considered. Under the influence of tectonic forces caused by mantle convection, the fault moves abruptly. The ensuing boundary value issues were solved using a numerical technique based on a finite element scheme with proper boundary conditions that was developed specifically for the task. The numerical algorithms described here can be tweaked to solve more general deformation issues where analytical methods become too complex. Appropriate modifications can also be incorporated in our model in case when the tectonic forces are not anti-symmetric in nature as well as fault plane is not vertical.

This work is devoted to the nonlinear inverse problem of identifying the reaction coefficient in an elliptic boundary value problem from single Cauchy data on a part of the boundary. We then examine simultaneously two elliptic boundary value problems generated from the available Cauchy data. The output least squares method with the Tikhonov regularization is applied to find approximations of the sought coefficient. We discretize the PDEs with piecewise linear finite elements. The stability and convergence of this technique are then established. A numerical experiment is presented to illustrate our theoretical findings., Reaction coefficient identification, Finite element method, Boundary observation, Tikhonov regularization, Neumann problem, Mixed problem, Ill-posed problem

The system { u t = Δ u − χ ∇ ⋅ ( u v ∇ v ) − u v + B 1 ( x , t ) , v t = Δ v + u v − v + B 2 ( x , t ) , ( ⋆ ) is considered in a disk Ω ⊂ R 2 , with a positive parameter χ and given nonnegative and suitably regular functions B 1 and B 2 defined on Ω × ( 0 , ∞ ) . In the particular version obtained when χ = 2 , (⋆) was proposed in [31] as a model for crime propagation in urban regions. Within a suitable generalized framework, it is shown that under mild assumptions on the parameter functions and the initial data the no-flux initial-boundary value problem for (⋆) possesses at least one global solution in the case when all model ingredients are radially symmetric with respect to the center of Ω. Moreover, under an additional hypothesis on stabilization of the given external source terms in both equations, these solutions are shown to approach the solution of an elliptic boundary value problem in an appropriate sense. The analysis is based on deriving a priori estimates for a family of approximate problems, in a first step achieving some spatially global but weak initial regularity information which in a series of spatially localized arguments is thereafter successively improved. To the best of our knowledge, this is the first result on global existence of solutions to the two-dimensional version of the full original system (⋆) for arbitrarily large values of χ.

We consider an elliptic boundary value problem with degenerate nonlocal term. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of zeros of the degenerate term. The solutions are also ordered according to their L p -norms.

In this paper we establish sufficient conditions for an asymptotically linear elliptic boundary value problem to have at least five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues. We make extensive use of variational methods and characterizations of the local degree of critical points. Furthermore, uniqueness and qualitative properties of the solutions are investigated.

Consider a second-order elliptic boundary value problem in three dimensions with locally smooth coefficients and solution. Discuss local superconvergence estimates for the tensor-product finite element approximation on a regular family of rectangular meshes. It will be shown that, by the estimates for the discrete Green’s function and discrete derivative Green’s function, and the relationship of norms in the finite element space such as L2 -norms, W1,∞ -norms, and negative-norms in locally smooth subsets of the domain Ω, locally pointwise superconvergence occurs in function values and derivatives.

An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum of the nonlinear problem under consideration is closed ( consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, . The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that and that for each the problem has a nontrivial semiregular solution. If there exists a positive constant such that the sum of the nonlinearity and is a function which is nondecreasing in the phase variable , then for any the boundary value problem has a minimal nontrivial solution . The required solution is semiregular, and is a decreasing mapping with respect to on . Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered. Bibliography: 37 titles.

We study a nonlinear critical second-order PDE with zero Dirichlet boundary condition. We prove existence and compactness results for the equation by using Bahri's method of critical points at infinity.

This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained by numerical integration. The theoretical results are verified by numerical experiments.

In this paper, the authors present a new condition of the uniqueness of the solution for a previous 1 D semi-linear elliptic boundary value problem of membrane MEMS devices, where the amplitude of the electric field is considered proportional to the curvature of the membrane. The existence of the solution (membrane deflection) depends on the material of the membrane, which is obtained by Schauder–Tychonoff’s fixed point approach. Thus, in this paper, the result of uniqueness has been completely reformulated to obtain a condition depending on the material of the membrane achieving a new result of existence and uniqueness, depending on both the material of the membrane and the geometrical characteristics of the device. Then, by shooting numerical method, more realistic conditions for detecting eventual ghost solutions and new ranges of both operational parameters and mechanical tension of the membrane ensuring convergence have been achieved confirming the useful information on the industrial applicability of the model under study.

In the paper [1], the geometrical mapping techniques based on Non-Uniform Rational B-Spline (NURBS) were introduced to solve an elliptic boundary value problem containing a singularity. In the mapping techniques, the inverse function of the NURBS geometrical mapping generates singular functions as well as smooth functions by an unconventional choice of control points. It means that the push-forward of the NURBS geometrical mapping that generates singular functions, becomes a piecewise smooth function. However, the mapping method proposed is not able to catch singularities emerging at multiple locations in a domain. Thus, we design the geometrical mapping that generates singular functions for each singular zone in the physical domain. In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in C0 space. Also, we modify the B-spline functions whose supports include the interface between/among them. We put the idea in practice by solving elliptic boundary value problems containing multiple singularities.

A mixed finite element method is developed for a semilinear fourth-order elliptic boundary value problem. The existence and uniqueness of the solutions to the mixing problem are proved by constructing an auxiliary problem. Moreover, a computational format is provided for numerical calculation, the convergence of this problem is proved under some hypothetical conditions. The optimal error estimate of the semilinear fourth-order elliptic boundary value problem is analyzed by a two-grid method. Finally, some numerical examples are provided to support the theoretical analysis.