scholarly journals Quasilinear Dirichlet problems with competing operators and convection

2020 ◽  
Vol 18 (1) ◽  
pp. 1510-1517
Author(s):  
Dumitru Motreanu
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximation Procedure ◽  
Convection Term ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Variational Structure

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

scholarly journals Existence of Solutions for the Evolution -Laplacian Equation Not in Divergence Form

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Changchun Liu ◽  
Junchao Gao ◽  
Songzhe Lian
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximate Solutions ◽  
Divergence Form ◽  
Existence Of Weak Solutions ◽  
Uniform Estimates ◽  
Laplacian Equation

The existence of weak solutions is studied to the initial Dirichlet problem of the equation , with inf . We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

Continuous solutions and approximating scheme for fractional Dirichlet problems on Lipschitz domains

2018 ◽  
Vol 149 (2) ◽  
pp. 533-560
Author(s):  
Patricio Felmer ◽  
Erwin Topp
Keyword(s):  
Dirichlet Problem ◽  
Approximation Scheme ◽  
Linear Operators ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Uniqueness Of Solutions ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Continuous Boundary ◽  
Extra Assumption

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

scholarly journals Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

2021 ◽  
Vol 66 (1) ◽  
pp. 95-103
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Integrable Function ◽  
Generalized Solution ◽  
Dirichlet Problems ◽  
Reaction Term ◽  
Fixed Point Approach

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem."

scholarly journals Some hemivariational inequalities in the Euclidean space

2019 ◽  
Vol 9 (1) ◽  
pp. 958-977 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Hemivariational Inequalities ◽  
Existence Of Weak Solutions ◽  
Lipschitz Continuous ◽  
Variational Structure ◽  
Locally Lipschitz ◽  
Sign Changing Solutions ◽  
Continuous Functionals

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

scholarly journals Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaohua He ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Yonglin Xu
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Order Term ◽  
Combined Effects ◽  
Convection Term ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Lower Order Terms

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

scholarly journals Existence results for some nonlinear degenerate problems in the anisotropic spaces

2021 ◽  
Vol 39 (6) ◽  
pp. 53-66
Author(s):  
Mohamed Boukhrij ◽  
Benali Aharrouch ◽  
Jaouad Bennouna ◽  
Ahmed Aberqi
Keyword(s):  
Elliptic Problem ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Sign Condition ◽  
Degenerate Elliptic ◽  
Anisotropic Spaces ◽  
Nonlinear Degenerate

Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,where for $i=1,...,N$ $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and $H_i(x,u,\nabla u)$ is a nonlinear term without a sign condition. Under suitable conditions on $a_i$ and $H_i$, we prove the existence of weak solutions.

On weak solutions to a dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases

2020 ◽  
Vol 30 (08) ◽  
pp. 1517-1553
Author(s):  
Young-Sam Kwon ◽  
Antonin Novotny ◽  
C. H. Arthur Cheng
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Open Problem ◽  
Type System ◽  
Convergent Subsequence ◽  
Original System ◽  
Heat Conducting ◽  
Existence Of Weak Solutions ◽  
Essential Step ◽  
Mathematical Literature

In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential step to the proof of the existence of weak solutions. This is the first result of this type in the mathematical literature. Nevertheless, the construction of weak solutions to this system however remains still an (difficult) open problem.

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

2019 ◽  
Vol 19 (1) ◽  
pp. 197-217 ◽  
Author(s):  
Caifeng Zhang
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Existence Of Weak Solutions ◽  
Laplacian Equation ◽  
Trudinger Inequality ◽  
Dimension One

Abstract In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in {W^{s,p}(\mathbb{R}^{N})} . Define \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where {s\in(0,1)} , {sp=N} , {\alpha\in[0,\alpha_{*})} and \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where {V(x)} has a positive lower bound, {f(x,t)} behaves like {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and {\varepsilon>0} . Moreover, we also derive a weak solution with negative energy.

QUASISTATIC EVOLUTION IN THE THEORY OF PERFECTLY ELASTO-PLASTIC PLATES PART I: EXISTENCE OF A WEAK SOLUTION

2009 ◽  
Vol 19 (02) ◽  
pp. 229-256 ◽  
Author(s):  
ALEXEY DEMYANOV
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Flow Rule ◽  
Variational Theory ◽  
Absolutely Continuous ◽  
Quasistatic Evolution ◽  
Existence Of Weak Solutions ◽  
Strong Formulation

The existence of weak solutions to the quasistatic problems in the theory of perfectly elasto-plastic plates is studied in the framework of the variational theory for rate-independent processes. Approximate solutions are constructed by means of incremental variational problems in spaces of functions with bounded hessian. The constructed weak solution is shown to be absolutely continuous in time. A strong formulation of the flow rule is obtained.

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