Laplace-type Equations Research Articles

Blow‐up for a nonlocal semilinear pseudo‐parabolic p$$ p $$‐Laplacian type equation

In this paper, a nonlocal semilinear pseudo‐parabolic ‐Laplacian type equation is considered, and finite time blow‐up of solution with different initial energy is proved. More precisely, the upper bound for the blow‐up time of solution with subcritical initial energy is established by potential well method and concavity argument. When the initial energy is supercritical, we prove the finite time blow‐up of solution and obtain the upper bound for the blow‐up time by different invariant set and concavity inequality. Moreover, by constructing a new control functional and providing careful estimates, the lower bound for the blow‐up time of solution is given.

Pairs of Positive Solutions for a Carrier p(x)-Laplacian Type Equation

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p(x)-Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč’s version of the celebrated Díaz–Saá’s inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each “positive bump” of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed.

Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

Abstract This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) r\left(x) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω \omega -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.

Sobolev capacities and trace inequalities related to $ \alpha- $Hermite operators with application to $ p $-Laplace type equations

Sobolev capacities and trace inequalities related to $ \alpha- $Hermite operators with application to $ p $-Laplace type equations

Anisotropic p-Laplace Equations on long cylindrical domain

The main aim of this article is to study the Poisson type problem for anisotropic \(p\)-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincar� inequality for a pseudo \(p\)-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincar� constant of an analogous problem set on a lower dimension.

Regularity for solutions of elliptic $$p(x)-$$Laplacian type equations with lower order terms and hardy potential

Regularity for solutions of elliptic $$p(x)-$$Laplacian type equations with lower order terms and hardy potential

Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p -Laplace equation -\Delta_{p} u=\mu with p\in (1,2) . The cases when p\in (2-\frac{1}{n},2) and p\in (\frac{3n-2}{2n-1},2-\frac{1}{n}] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379–418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p\in(1,\frac{3n-2}{2n-1}] . Interior and global modulus of continuity estimates of the gradients of solutions are also established.

Regularity results for (p,q)-Laplace type equations in generalized Morrey spaces

Regularity results for (p,q)-Laplace type equations in generalized Morrey spaces

Solution of fractional Laplace type equation in conformable sense using fractional fourier series with separation of variables technique

Solution of fractional Laplace type equation in conformable sense using fractional fourier series with separation of variables technique

Rigidity results for the p‐Laplace type equations on compact Riemannian manifolds

In this paper, we obtain two rigidity results for ‐Laplace type equation and ‐Laplace equation with exponential nonlinearity on ‐dimensional compact Riemannian manifolds by using of nonlinear flow and the carré du champ methods, respectively, where rigidity means that the PDE has only constant solution when a parameter is in a certain range. Moreover, an interpolation inequality is derived as an application.

The inhomogeneous boundary Harnack principle for fully nonlinear and $p$-Laplace equations

We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and p -Laplace-type equations with a right-hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.

NUMERICAL SOLUTIONS FOR 2D UNSTEADY LAPLACE-TYPE PROBLEMS OF ANISOTROPIC FUNCTIONALLY GRADED MATERIALS

The time-dependent Laplace-type equation of variable coefficients for anisotropic inhomogeneous media is discussed in this paper. Numerical solutions to problems which are governed by the equation are sought by using a combined Laplace transform and boundary element method. The variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation after being Laplace transformed is then written in a boundary-only integral equation involving a time-free fundamental solution. The boundary integral equation is therefore employed to find the numerical solutions using a standard boundary element method. Finally the numerical results are inversely transformed numerically using the Stehfest formula to obtain solutions in the time variable. Some problems of anisotropic functionally graded media are considered. The results show that the combined Laplace transform and boundary element method is accurate and easy to implement.

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

Linear Integrable Systems on Quad-Graphs

Abstract In the first part of the paper, we classify linear integrable (multidimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb{C}$, enjoying natural symmetries and the property that the restriction of their solutions to the black vertices satisfies a Laplace type equation. The classification reduces to solving a functional equation. Under certain restriction, we give a complete solution of the functional equation, which is expressed in terms of elliptic functions. We find two real analytic reductions, corresponding to the cases when the underlying complex torus is of a rectangular type or of a rhombic type. The solution corresponding to the rectangular type was previously found by Boutillier, de Tilière, and Raschel. Using the multidimensional consistency, we construct the discrete exponential function, which serves as a basis of solutions of the quad-equation. In the second part of the paper, we focus on the integrability of discrete linear variational problems. We consider discrete pluri-harmonic functions, corresponding to a discrete two-form with a quadratic dependence on the fields at black vertices only. In an important particular case, we show that the problem reduces to a two-field generalization of the classical star-triangle map. We prove the integrability of this novel 3D system by showing its multidimensional consistency. The Laplacians from the first part come as a special solution of the two-field star-triangle map.

Uniqueness of positive solutions for boundary value problems associated with indefiniteϕ-Laplacian-type equations

AbstractThis paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with theϕ-Laplacian equation(ϕ(u′))′+a(t)g(u)=0,(\phi \left(u^{\prime} ))^{\prime} +a\left(t)g\left(u)=0,whereϕis a homeomorphism withϕ(0) = 0,a(t) is a stepwise indefinite weight andg(u) is a continuous function. When dealing with thep-Laplacian differential operatorϕ(s) = ∣s∣p−2swithp > 1, and the nonlinear termg(u) = uγwithγ∈R\gamma \in {\mathbb{R}}, we prove the existence of a unique positive solution whenγ ∈ ]−∞\infty, (1 − 2p)/(p − 1)] ∪ ]p − 1, +∞\infty[.

Heat and Laplace type equations with complex spatial variables in weighted Fock spaces

In a recent book co-authored by the authors of this article, we studied by semigroup theory methods several classical evolution equations, including the heat and Laplace equations, with real time variable and complex spatial variable, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. Also, in a subsequent paper, the authors have extended the results for the heat and Laplace equations in weighted Bergman spaces on the unit disk. The purpose of this article is to show that the semigroup theory methods work for these two evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the weighted Fock space on \(\mathbb{C}\), \(F_{\alpha }^p(\mathbb{C})\), with \(1\leq p<+\infty \), endowed with The \(L^p\)-norm. Also, the case of several complex variables is considered. The proofs use the Jensen's inequality, Fubini's theorem for integrals and the \(L^p\)-integral modulus of continuity.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/109/abstr.html

$1/8$-BPS couplings and exceptional automorphic functions

Unlike the \mathcal{R}^4ℛ4 and \nabla^4\mathcal{R}^4∇4ℛ4 couplings, whose coefficients are Langlands–Eisenstein series of the U-duality group, the coefficient \mathcal{E}^{(d)}_{(0,1)}ℰ(0,1)(d) of the \nabla^6\mathcal{R}^4∇6ℛ4 interaction in the low-energy effective action of type II strings compactified on a torus T^dTd belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work [1], it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states running in the loops, up to the addition of a particular Langlands–Eisenstein series. Here we compute the weak coupling and large radius expansions of these automorphic functions for any dd. We find perfect agreement with perturbative string theory up to genus three, along with non-perturbative corrections which have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS instantons and anti-instantons. The additional Langlands–Eisenstein series arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS states running in the loops, and the three-loop amplitude with mutually 1/2-BPS states in the loops. For d=4d=4, the result is shown to coincide with an alternative proposal [2] in terms of a covariantised genus-two string amplitude, due to interesting identities between the Kawazumi–Zhang invariant of genus-two curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest.

A continuous time tug-of-war game for parabolic p(x,t)-Laplace-type equations

We formulate a stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized [Formula: see text]-Laplace operator. Our game is formulated in a way that covers the full range [Formula: see text]. Furthermore, we prove the uniqueness of viscosity solutions to our equation in the whole space under suitable assumptions.

Emanating Jets As Shaped by Surface Tension Forces.

We show that emanating jets can be regarded as growing liquid towers, which are shaped by the twofold action of surface tension: first the emanated fluid is being accelerated back by surface tension force, herewith creating the boundary conditions to solve the shape of the liquid tower as a solution of an equation mathematically related to the hydrostatic Young−Laplace equation, known to give solutions for the shape of pending and sessile droplets, and wherein the only relevant forces are gravity g and surface tension γ. We explain that for an emanating jet under specific constraints all mass parts with density ρ will experience a uniform time dependent acceleration a(t). An asymptotic solution is subsequently numerically derived by making the corresponding Young−Laplace type equation dimensionless and by dividing all lengths by a generalized time dependent capillary length λc(t) = . The time dependent surface tension γ(t) can be derived by measuring both time dependent acceleration a(t) and time dependent capillary length λc(t). Jetting experiments with water and coffee show that the dynamic surface tension behavior according to the emanating jet method and with the well-known maximum bubble pressure method are the same, herewith verifying the proposed model.

Multiplicity of solutions to discrete inclusions with the p(k)-Laplace type equations

AbstractIn this article, we prove the existence and multiplicity of solutions to discrete inclusions with the p(k)-Laplace type equations. We are interested in the existence of three solutions with the aid of linking arguments and using a three critical points theorem, for locally Lipschitz continuous fonctions.