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This article introduces a finite element method based on the C-Bézier basis function for second-order elliptic equations. The trial function of the finite element method is set up using a combination of C-Bézier tensor product bases. One advantage of the C-Bézier basis is that it has a free shape parameter, which makes geometric modeling more convenience and flexible. The performance of the C-Bézier basis is searched for by studying three test examples. The numerical results demonstrate that this method is able to provide more accurate numerical approximations than the classical Lagrange basis. [ABSTRACT FROM AUTHOR]

We consider the equation − Δ w = w 3 with zero boundary conditions on planar domains that are conformal images of annuli. Starting with an approximate solution, we prove that there exists a true solution nearby. Our approach is computer-assisted. It involves simultaneous and accurate control of the (inverse) Dirichlet Laplacean, nonlinearities, and conformal mappings. [ABSTRACT FROM AUTHOR]

We solve the inverse Cauchy problem of elliptic type partial differential equations in an arbitrary 3D closed walled shell for recovering unknown data on an inner surface, with the over-specified Cauchy boundary conditions given on an outer surface. We first derive a homogenization function in the 3D domain to annihilate the Dirichlet as well as the Neumann data on the outer surface. Then, we can transform the inverse Cauchy problem to solve a direct problem inside the closed walled shell, using the homogenization technique and the domain type collocation method. The boundary functions are constructed from the 3D Pascal polynomials multiplied by an elementary boundary function, which are adopted as the bases to expand the numerical solution of the transformed elliptic equation. A simple scaling regularization is employed to reduce the condition number of the linear system to determine the expansion coefficients. Several numerical examples are presented to show that the novel method can overcome the highly ill-posed property of the inverse Cauchy problem in the 3D closed walled shell. The proposed algorithm is robust against large noise up to 20 % and very time saving to obtain accurate solution. [ABSTRACT FROM AUTHOR]

We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation , where the coefficient is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to , , and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy. [ABSTRACT FROM AUTHOR]

Abstract We prove a global Calderón–Zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right-hand side is a bounded Radon measure. Minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate. [ABSTRACT FROM AUTHOR]

Abstract In this work we consider the existence of positive solutions to various equations of the form − Δ u (x) = (1 + g (| x | , u)) u (x) p in B R , u = 0 on ∂ B R , where B R is the open ball of radius R in R N centered at the origin and p = N + 2 N − 2. We will generally assume g is nonnegative. Our approach will be to utilize some dynamical systems approaches. [ABSTRACT FROM AUTHOR]

Abstract We study the following coupled system of quasilinear equations: − Δ p u + μ | u | p − 2 u = | u | q − 2 u + α λ | u | α − 2 u | v | β , x ∈ R N , − Δ p v + ν | v | p − 2 v = | v | p ∗ − 2 v + β λ | u | α | v | β − 2 v , x ∈ R N , where N ≥ 3 , μ , ν , λ > 0 , α , β ≥ 1 , 2 < p < q < p ∗ and α + β = p. We establish some results about the existence and regularity of ground state solutions for the p-Laplacian systems by using variational methods. We also study the asymptotic behavior of solutions as the coupling parameter λ tends to zero. [ABSTRACT FROM AUTHOR]

Abstract Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained by symmetrization techniques. The anisotropy of the principal part of the operator is governed by a general n -dimensional Young function of the gradient which is not necessarily of polynomial type and need not satisfy the Δ 2 -condition. [ABSTRACT FROM AUTHOR]

Abstract A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the quasilinear elliptic problem − div ∇ u ∕ 1 + | ∇ u | 2 + a (x) u = b (x) ∕ 1 + | ∇ u | 2 in B , u = 0 on ∂ B , where B is an open ball in R N , with N ≥ 2 , and a , b ∈ C 1 (B ¯) are given radially symmetric functions, with a (x) ≥ 0 in B. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a , b are not necessarily constant and no sign condition is assumed on b. [ABSTRACT FROM AUTHOR]

Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it has been known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Gutiérrez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting. [ABSTRACT FROM AUTHOR]

Abstract Let Ω ⊂ R N (N ≥ 3) be a bounded C 2 domain and δ (x) = dist (x , ∂ Ω). Put L μ = Δ + μ δ 2 with μ > 0. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to − L μ u = u p + τ in Ω , u = ν on ∂ Ω , where μ > 0 , p > 0 , τ and ν are measures on Ω and ∂Ω respectively. We then establish existence results for the system { − L μ u = ϵ v p + τ in Ω , − L μ v = ϵ u p ˜ + τ ˜ in Ω , u = ν , v = ν ˜ on ∂ Ω , where ϵ = ± 1 , p > 0 , p ˜ > 0 , τ and τ ˜ are measures on Ω, ν and ν ˜ are measures on ∂Ω. We also deal with elliptic systems where the nonlinearities are more general. [ABSTRACT FROM AUTHOR]

Abstract A revised Yau's Curvature Difference Flow is considered to deform one convex curve X 0 to another one X ˜. It is proved that this flow exists globally on time interval [ 0 , + ∞) and the evolving curve, preserving its convexity and bounded area A , converges to a fixed limiting curve X ∞ (congruent to A / A ˜ X ˜) as time tends to infinity, where A ˜ is the area bounded by the target curve X ˜. [ABSTRACT FROM AUTHOR]

In this paper, we establish a global compactness result for (P.S.) sequences of the variational functional of the elliptic problem − Δ u − μ | x | 2 u = 1 | x | s | u | 2 s ∗ − 2 u + λ u , x ∈ Ω , u = 0 on ∂ Ω , where Ω ⊂ R n , n ≥ 3 , is a bounded smooth domain with 0 ∈ Ω , μ ∈ [ 0 , ( n − 2 ) 2 ∕ 4 ) , s ∈ [ 0 , 2 ) and λ ∈ R are constants. This extends the global compactness result of Cao and Peng (2003) to the case of elliptic problems with double singular critical terms. Our arguments adapt some refined Sobolev inequalities systematically developed quite recently by Palatucci and Pisante (2014) and blow-up analysis. In this way, our arguments turn out to be quite transparent and easy to be applied to many other problems. [ABSTRACT FROM AUTHOR]

This article is concerned with the analysis of the discontinuous Galerkin method (DGM) for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The growth of the nonlinearity is not compatible with the differential equation, which represents an obstacle in the analysis of the problem. Using monotone operator theory, it is possible to prove the existence and uniqueness of the weak solution and the approximate DG solution. The main emphasis is on the study of error estimates. To this end, the regularity of the weak solution is investigated, and it is shown that due to the singular boundary points, the solution loses regularity in the vicinity of these points. It transpires that the error estimation depends essentially on the opening angle of the corner points and the nonlinearity in the boundary term. It also depends on the parameter defining the nonlinear behaviour of the Newton boundary condition. At the end of this article, some computational experiments are presented. [ABSTRACT FROM AUTHOR]

Abstract We study nonlinear elliptic equations of p -Laplacian type in divergence form to establish a natural Calderón–Zygmund type theory of an Orlicz space type, where the Lebesgue space is the special case with Φ (t) = t q p : Φ (| F | p) ∈ L 1 (Q 2 R) ⟹ Φ (| D u | p) ∈ L 1 (Q R). In the previous results, the estimates obtained were strictly above the natural exponents, and the function such as Φ (t) = t log (1 + t) was ruled out for the candidate of Φ. But with our approach, Φ can be selected up to the end point case of the estimates, and the functions Φ (t) = t and Φ (t) = t log (1 + t) can be additionally selected for Φ. [ABSTRACT FROM AUTHOR]

Abstract We will browse a series of results on elliptic equations and systems, with particular nonlinear lower order terms and measure right-hand side, obtained in the last years in the papers (Cirmi and Leonardi, 2014; Cianci et al., 2017; Cirmi et al., 2014, 2017, 2018). Namely we take into account equations whose prototypes are: − Δ u + u | D u | 2 = f or − Δ u + | D u | 2 u θ = f , θ ∈ ] 0 , 1 [ , where the right-hand side f belongs to a suitable Morrey space, for instance to L 1 , λ , 0 < λ ≤ N − 2 , and we prove corresponding Morrey estimates for the gradient of a solution. [ABSTRACT FROM AUTHOR]

Abstract In this article we are interested in the nonlinear Schrödinger equation with nonlocal regional diffusion: (0.1) (− Δ) ρ α u + u = f (x , u) in R n , u ∈ H α (R n) , where α ∈ (0 , 1) and (− Δ) ρ α is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρ (x) > 0. By using the symmetric mountain pass theorem and the genus properties in critical point theory, we show that problem (0.1) has infinitely many solutions. Recent results in the literature are significantly improved. [ABSTRACT FROM AUTHOR]

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f  in  Ω,  u=0  on  ∂Ω,}, where 00 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation. [ABSTRACT FROM AUTHOR]

Abstract The paper deals with a nonlinear nonlocal second-order p -Laplacian equation with Dirichlet boundary conditions. A rigorous proof for existence and uniqueness of the weak solution is presented. The weak formulation of the problem of interest is transformed into an unconstrained minimization problem. A variational inequality for the objective functional is obtained. A priori estimates for the weak solution are proved. The optimization problem is solved by means of the two-point step size gradient method. A steplength for the iterative method assuring monotone decrease of the error in approximate solutions is found. Q -linear convergence of the finite element approximations to the discrete solution is established. A monotone error reduction is proved. An original procedure for obtaining initial guesses is developed. Numerical examples supporting the developed theory are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we derive Schauder estimates to the solutions of the following uniformly elliptic equation with a inverse-square potential and nonhomogeneous term (0.1) − a i j ( x ) ∂ i j u ( x ) + A | x | 2 u ( x ) = f ( x ) , in B , which lead to the existence and sharp regularity results of the classical solutions. More precisely, we prove that u ∈ C n + 2 , γ provided f ∈ C n , γ , a i j ∈ C n , γ and A > Λ ( 2 + n + γ ) ( d + n + γ ) . [ABSTRACT FROM AUTHOR]

This paper is concerned with an evolution problem having an elliptic equation involving the 1-Laplacian operator and a dynamical boundary condition. We apply nonlinear semigroup theory to obtain existence and uniqueness results as well as a comparison principle. Our main theorem shows that the solution we found is actually a strong solution. We also compare solutions with different data. [ABSTRACT FROM AUTHOR]

Abstract A version of Liouville's theorem is proved for solutions of some degenerate elliptic equations defined in R n \ K , where K is a compact set, provided the structure of this equation and the dimension n are related. Theoretical and numerical examples are given. Furthermore, a comparison result and the uniqueness of solution are obtained for such equations in exterior domains. [ABSTRACT FROM AUTHOR]

In this paper we develop an adaptive finite element method for elliptic problems. First, we assume that in each subdomain the norm of the approximation error at the current mesh configuration is bounded by the norm of the approximation error obtained at the previous mesh configuration, for some norm H s . Then an a-posteriori error estimator is designed based on the approximate solution correction between the solution on the last two mesh configurations. Based on this new error estimator, the element-wise refinement strategy in each subdomain is provided for a given tolerance. A discussion on the choice of the coefficients in the assumption is given for different norm spaces and for different degrees of finite element family. Four 2D numerical benchmark examples of different domains and two 3D numerical benchmark examples are tested to demonstrate the robustness of our method. When possible, our numerical results are also compared to corresponding results from existing methods. All the results show that the proposed method is robust and efficient in terms of the number of degrees of freedom. [ABSTRACT FROM AUTHOR]

The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach. [ABSTRACT FROM AUTHOR]

We construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension n = 4 , 5 . Such a construction is already available in the literature in dimensions n ≥ 6 (see for instance [10,14,30,32,36] ) and not possible in dimension 3 by [25] . This also provides new patterns for the Lin–Ni [26] problem on closed manifolds and completes results by Brézis and Li [8] about this problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish existence and multiplicity of solutions for the following class of quasilinear field equation    −Δu + V(x)u − Δpu + W′(u) = 0,  in  ℝN,    (P) where u = (u1, u2, ... , uN+1), p > N ≥ 2, W is a singular function and V is a positive continuous function. [ABSTRACT FROM AUTHOR]

In this paper, we study the solutions of elliptic equations involving regional fractional Laplacian (E) (− Δ) Ω α u = f in a bounded regular domain Ω in R N (N ≥ 2) with C2 boundary ∂Ω, subject to Dirichlet boundary g on ∂Ω, where α ∈ ( 1 2 , 1) and the operator (− Δ) Ω α denotes the regional fractional Laplacian. We prove that when g ≡ 0, problem (E) admits a unique weak solution under the hypotheses that f ∈ L2(Ω), f ∈ L1(Ω, ρβdx), and f ∈ M (Ω , ρ β ) , where ρ(x) = dist(x, ∂Ω), β = 2α – 1, and M (Ω , ρ β ) is a space of all Radon measures ν satisfying ∫Ωρβd|ν| < + ∞. Finally, we provide an integration by parts formula for the classical solution of (E) with boundary data g. [ABSTRACT FROM AUTHOR]

This paper is considered with the second expansion of the unique vanishing at infinity solution to − Δ u = b ( x ) g ( u ) , x ∈ R N ( N ≥ 3 ) , where b ∈ C ( R N ) is nonnegative and may be supercritical attenuation or critical attenuation at infinity, g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) is non-increasing on ( 0 , ∞ ) with lim s → 0 + g ( s ) = ∞ and g is normalized regularly varying at zero with index − γ ( γ > 1 ) . [ABSTRACT FROM AUTHOR]

The Heisenberg ferromagnetic spin chain equation is investigated. By applying the improved F‐expansion method (Exp‐function method) and the Jacobi elliptic method, respectively, a series of exact solutions is constructed. The parametric conditions of the existence for the solutions are presented. These solutions comprise periodic wave solutions, doubly periodic wave solutions, and dark and bright soliton solutions, which are expressed in several different function forms, namely, Jacobi elliptic function, trigonometric function, hyperbolic function, and exponential function. The results illustrate that the Exp‐function method is a powerful symbolic algorithm to look for new solutions for the nonlinear evolution systems. [ABSTRACT FROM AUTHOR]

This paper focuses on a nonhomogeneous quasilinear elliptic boundary blow up problem △ p u = a ( x ) f ( u ) + h ( x ) in Ω , u | ∂ Ω = ∞ , where p > 1 , Ω is a smooth bounded domain in R n , and h ∈ C ( Ω ) may be sign-changing in Ω and singular on ∂ Ω . We mainly analyze the influences caused by h on the existence of large solutions. Furthermore, it can be verified that any large solution is nonnegative under an additional assumption. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise Cα, C1,α and C2,α regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations. [ABSTRACT FROM AUTHOR]

This paper presents two discrete maximum principles (DMP) for the numerical solution of second order elliptic equations arising from the weak Galerkin finite element method. The results are established by assuming an h -acute angle condition for the underlying finite element triangulations. The mathematical theory is based on the well-known De Giorgi technique adapted in the finite element context. Some numerical results are reported to validate the theory of DMP. [ABSTRACT FROM AUTHOR]

We consider quasilinear equations of the p-Laplacian-type uniformly degenerating in a part of the domain. An analog of the Harnack inequality is established for the nonnegative solutions and the Hölder continuity of the solutions is proved by using this inequality. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation 1 ( − Δ ) α u = u p in Ω ∖ { 0 } , ( − Δ ) α u = 0 in R N ∖ Ω ,where p > 1, α ∈ ( 0 , 1 ), Ω is a bounded C 2 domain in R N containing the origin, N ⩾ 2 α and the fractional Laplacian ( − Δ ) α is defined in the principle value sense. We prove that any classical solution u of is a very weak solution of 2 ( − Δ ) α u = u p + k δ 0 in Ω , ( − Δ ) α u = 0 in R N ∖ Ωfor some k ⩾ 0, where δ 0 is the Dirac mass at the origin. In particular, when p ⩾ N N − 2 α, we have that k = 0; when p ∈ ( 1 , N N − 2 α ), u has removable singularity at the origin if k = 0 and if k > 0, u satisfies that lim x → 0 u ( x ) | x | N − 2 α = c N , α k ,where c N , α > 0. Furthermore, when p ∈ ( 1 , N N − 2 α ), we show that there exists k ∗ > 0 such that problem has at least two solutions for k ∈ ( 0 , k ∗ ), a unique solution for k = k ∗ and no solution for k > k ∗. [ABSTRACT FROM AUTHOR]

We consider the dynamics of an infinitesimal particle under the gravitational action of P primaries of equal masses. These move in an elliptic homographic solution of the P-body problem and the infinitesimal particle moves along the straight line perpendicular to their plane of motion and passing through the common focus of the ellipses. In this work we consider the parametric stability of the infinitesimal mass located at the focus of the ellipses. We construct the boundary curves of the stability/instability regions in the plane of the parameters μ and ϵ, which are the mass of each primary and the eccentricity of the elliptic orbit, respectively. [ABSTRACT FROM AUTHOR]

This paper deals with the homogenization problem of one-dimensional pseudo-elliptic equations with a rapidly varying random potential.The main purpose is to characterize the homogenization error (random fluctuations), i.e., the difference between the random solution and the homogenized solution, which strongly depends on the autocovariance property of the underlying random potential.It is well known that when the random potential has short-range dependence, the rescaled homogenization error converges in distribution to a stochastic integral with respect to standard Brownian motion.Here, we are interested in potentials with long-range dependence and we prove convergence to stochastic integrals with respect to the Hermite process. [ABSTRACT FROM AUTHOR]

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L2 space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ→∞ to the strongly collisional regime τ→0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ-dependent constraint on δ ensuring exponential decay with explicit τ-dependent rates towards the stationary solution. In the strongly collisional limit τ→0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L2 space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates. [ABSTRACT FROM AUTHOR]

In this article we obtain positive singular solutions of (1) − Δ u = | ∇ u | p in Ω , u = 0 on ∂ Ω , where Ω is a small C 2 perturbation of the unit ball in R N . For N N − 1 < p < 2 we prove that if Ω is a sufficiently small C 2 perturbation of the unit ball there exists a singular positive weak solution u of (1) . In the case of p > 2 we prove a similar result but now the positive weak solution u is contained in C 0 , p − 2 p − 1 ( Ω ‾ ) and yet is not in C 0 , p − 2 p − 1 + ε ( Ω ‾ ) for any ε > 0 . [ABSTRACT FROM AUTHOR]

In this paper, spreading properties of a competition-diffusion system of two equations are studied. This system models the invasion of an empty favorable habitat, by two competing species, each obeying a logistic growth equation, such that any coexistence state is unstable. If the two species are initially absent from the right half-line x > 0 , and the slowest one dominates the fastest one on x < 0 , then the latter will invade the right space at its Fisher-KPP speed, and will be replaced by or will invade the former, depending on the parameters, at a slower speed. Thus, the system forms a propagating terrace, linking an unstable state to two consecutive stable states. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term where,,is continuous. Under the conditions that the nonlinearitymay be of superlinear or sublinear growth inuand, existence results of positive radial solutions are obtained. For the superlinear case, the growth offinis restricted to quadratic growth. The superlinear and the sublinear growth of the nonlinearity offare described by inequality conditions instead of the usual upper and lower limits conditions. Our discussion is based on the fixed point index theory in cones. [ABSTRACT FROM AUTHOR]

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ . The associated Cameron–Martin space is denoted by H . Consider two sufficiently regular convex functions U : X → R and G : X → R . We let ν = e − U μ and Ω = G − 1 ( − ∞ , 0 ] . In this paper we are interested in the W 2 , 2 regularity of the weak solutions of elliptic equations of the type (0.1) λ u − L ν , Ω u = f , where λ > 0 , f ∈ L 2 ( Ω , ν ) and L ν , Ω is the self-adjoint operator associated with the quadratic form ( ψ , φ ) ↦ ∫ Ω 〈 ∇ H ψ , ∇ H φ 〉 H d ν ψ , φ ∈ W 1 , 2 ( Ω , ν ) . In addition we will show that if u is a weak solution of problem (0.1) then it satisfies a Neumann type condition at the boundary, namely for ρ -a.e. x ∈ G − 1 ( 0 ) 〈 Tr ( ∇ H u ) ( x ) , Tr ( ∇ H G ) ( x ) 〉 H = 0 , where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and Tr is the trace operator. [ABSTRACT FROM AUTHOR]

We prove a comparison principle for unbounded weak sub/super solutions of the equation λ u − div ( A ( x ) D u ) = H ( x , D u ) in Ω where A ( x ) is a bounded coercive matrix with measurable ingredients, λ ≥ 0 and ξ ↦ H ( x , ξ ) has a super linear growth and is convex at infinity. We improve earlier results where the convexity of H ( x , ⋅ ) was required to hold globally. [ABSTRACT FROM AUTHOR]

We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind are relevant for applications in engineering, chemistry and in biology, e.g., biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretization in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the proposed adaptive algorithm with several benchmark examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the entire blow-up solutions for a quasilinear p -Laplacian Schrödinger elliptic equation with a non-square diffusion term. By using the dual approach and some new iterative techniques, the difficulty due to the non-square diffusion term and the p -Laplacian operator is overcome and the nonexistence and existence of entire blow-up solutions are established. [ABSTRACT FROM AUTHOR]