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We study the existence of weak solutions for a system and a coupled system of Λ -Hilfer fractional differential equations on compact domains using the Lax–Milgram and Minty–Browder theorems. Furthermore, we provide an illustrative example, and a regularity result to imply that the obtained solution is classical. [ABSTRACT FROM AUTHOR]

In this work, a known Eddy Current (EC) model is adapted to characterize subsurface defects in carbon fiber-reinforced polymer (CFRP) plates intended for the civil aerospace industry. The considered defects include delaminations, microcracks, porosity, fiber breakage, and the simultaneous presence of these defects. Each defect is modeled as an additive variation in the material's electrical conductivity tensor, allowing for a detailed mathematical representation of the defect's influence on the CFRP's electromagnetic behavior. The additivity of the variations in the conductivity tensor is justified by the assumption that the defects are not visible to the naked eye, implying that the material does not require non-destructive testing. The adapted EC model admits a unique and stable solution by verifying that all analytical steps are satisfied. To reconstruct 2D maps of the magnetic flux density amplitude, a FEM formulation is adopted, based on the energy functional because it ensures a stable and consistent numerical formulation given its coercivity. Moreover, the numerical approach allows precise and reliable numerical solutions, enhancing the capability to detect and quantify defects. The numerical results show that the obtained 2D maps are entirely superimposable on those highlighting the distribution of mechanical stress states known in the literature, offering a clear advantage in terms of detection costs. This approach provides an effective and economical solution for the non-destructive inspection of CFRP, ensuring accurate and timely defect diagnosis for maintaining structural integrity. [ABSTRACT FROM AUTHOR]

In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

In this paper, the existence of solutions for a second-order impulsive differential equation with a parameter on the half-line is investigated. Applying Lax-Milgram Theorem, we deal with a linear Dirichlet impulsive problem, while the nonlinear case is established by using standard results of critical point theory.

In the present paper, we investigate the existence and regularity of weak solutions for ψ -Hilfer fractional boundary value problem in C 2 α , β ; ψ and H (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for ψ -Riemann–Liouville fractional integral and ψ -Hilfer fractional derivative. [ABSTRACT FROM AUTHOR]

A vector problem of electromagnetic-wave diffraction by a cylinder is described by a system of two two-dimensional integro-differential equations. After expanding the unknown functions and the right-hand sides in Fourier series, the problem reduces to systems of one-dimensional equations. Analytical inversion of the principal operator of one-dimensional systems in Sobolev spaces is considered. Theorems on the boundedness and bounded invertibility of the principal operator are proved. The inverse operator is represented by series and in closed form: the elements of the inverse matrix are integral or integro-differential operators. [ABSTRACT FROM AUTHOR]

We present three versions of the Lax–Milgram theorem in the framework of Hilbert C ∗ -modules, two for self-dual ones over W ∗ -algebras and one for those over C ∗ -algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert C ∗ -modules over C ∗ -algebras of compact operators, however, the modular Lax–Milgram theorem turns out to be valid for all of them. We also give several examples to illustrate our results, in particular, we show that the main theorem is not true for Hilbert modules over arbitrary C ∗ -algebras. [ABSTRACT FROM AUTHOR]

In this article, we present a biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. The tumor tissue considered here is an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are wet by a physiological extracellular fluid. Since the tumor is deformable in nature, the mass and momentum equations for both the phases are presented. The momentum equations are coupled due to the interaction (or drag) force term. These governing equations reduce to a one-way coupled system under the assumption of infinitesimal deformation of the solid phase. The well-posedness of this model is shown in the weak sense by using the inf-sup (Babuska–Brezzi) condition and Lax–Milgram theorem in 2D and 3D. Further, we discuss a one-dimensional spherical symmetry model and present some results on the stress fields and energy of the system based on 𝐿2 and Sobolev norms. We discuss the so-called phenomena of “necrosis” inside a solid tumor using the energy of the system.

In this article, using an iterative technique, we introduce an extension of the Lax-Milgram theorem which can be used for proving the existence of solutions to boundary-value problems. Also, we apply of the obtained result to the fractional differential equation $$\displaylines{ {}_t D_T^{\alpha}{}_0 D_t^{\alpha}u(t)+u(t) =\lambda f (t, u(t)) \quad t \in (0,T),\cr u(0)=u(T)=0, }$$ where ${}_tD_T^\alpha$ and ${}_0D_t^\alpha$ are the right and left Riemann-Liouville fractional derivative of order $\frac{1}{2}< \alpha \leq 1$ respectively, $\lambda$ is a parameter and $f:[0,T]\times\mathbb{R}\to\mathbb{R}$ is a continuous function. Applying a regularity argument to this equation, we show that every weak solution is a classical solution.

For bounded domains Ω , we prove that the Lp -norm of a regular function with compact support is controlled by weighted Lp -norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set , where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp -based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic , degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational : the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem. [ABSTRACT FROM AUTHOR]

In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax–Milgram Theorem. [ABSTRACT FROM AUTHOR]

In this note we introduce the concept of a weak solution for a linear equation with not instantaneous impulses. We use the classical Lax–Milgram Theorem to reveal the variational structure of the problem and get the existence and uniqueness of weak solutions as critical points. This will allow us in the future to deal with the corresponding nonlinear problems. [ABSTRACT FROM AUTHOR]

In this article, using an iterative technique, we introduce an extension of the Lax-Milgram theorem which can be used for proving the existence of solutions to boundary-value problems. Also, we apply of the obtained result to the fractional differential equation tDΤα 0Dtα u(t) = u(t) + λƒ (t, u(t) t ∈ (0, Τ), u(0) = u(Τ) = 0, where tD Τα and 0Dtα are the right and left Riemann-Liouville fractional derivative of order ½ < α ≤ 1 respectively, λ is a parameter and ƒ:[0, Τ] x ℝ → ℝ is a continuous function. Applying a regularity argument to this equation, we show that every weak solution is a classical solution. [ABSTRACT FROM AUTHOR]

We develop the Galerkin method for a recent version of the Lax–Milgram theorem. The generation of the corresponding finite-dimensional subspaces for concrete boundary value problems leads us to consider certain biorthogonal systems in the reflexive Banach spaces in question. In addition, we present an application to the numerical solution of inverse problems involving certain elliptic boundary value problems. [ABSTRACT FROM AUTHOR]

Abstract: By adopting some ideas of the theory of functional inequalities we obtain a modification of the Lax–Milgram theorem. We drop linearity assumptions by assuming that respective inequalities hold true and we impose only a mild regularity assumption. Moreover, we do not assume the coercivity, however, as a consequence, our assertions are adequately weaker. One of our tools is a selection theorem for multifunctions. [Copyright &y& Elsevier]

Abstract: In the paper, we consider the existence of the solution of the second-order impulsive differential equations with inconstant coefficients. We change the second-order impulsive partial differential equation into the equivalent equation by transformation. By using the critical point theory of variational method and Lax-Milgram theorem, we obtain new results for the existence of the solution of the impulsive partial differential equations. [Copyright &y& Elsevier]

Abstract: In this work we deal with a constrained variational equation associated with the usual weak formulation of an elliptic boundary value problem in the context of Banach spaces, which generalizes the classical results of existence and uniqueness. Furthermore, we give a precise estimation of the norm of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence and multiplicity of solutions for some Dirichlet impulsive problems and some existence and multiplicity results are obtained. The solutions are sought by means of Lax–Milgram theorem and some critical theorems. [Copyright &y& Elsevier]

Abstract: Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work we present a new approach via variational methods and critical point theory to obtain the existence of solutions to impulsive problems. We consider a linear Dirichlet problem and the solutions are found as critical points of a functional. We also study the nonlinear Dirichlet impulsive problem. [Copyright &y& Elsevier]

The problem of representing linear functionals and linear operators using a given bilinear mapping is considered. For linear continuous functionals and linear compact operators that act in Banach and locally convex spaces, theorems on their representability using a given bilinear operator are formulated. Our results generalize the classical Lax-Milgram theorem. [ABSTRACT FROM AUTHOR]

Abstract: We obtain the nonlinear versions of the Stampacchia theorem and the Lax–Milgram theorem. Our results are stronger than the classical ones even in the linear case. Applying these theorems we get nontrivial solutions of -Laplace elliptic and pseudo--Laplace problems. [Copyright &y& Elsevier]

Existence and uniqueness of the solution of a non-stationary transport equation subject to the boundary and the initial conditions are proved via the Hille-Yosida theory. [ABSTRACT FROM AUTHOR]

In this paper the linear gravimetric boundary-value problem is discussed in the sense of the so-called weak solution. For this purpose a Sobolev weight space was constructed for an unbounded domain representing the exterior of the Earth and quantitative estimates were deduced for the trace theorem and equivalent norms. In the generalized formulation of the problem a special decomposition of the Laplace operator was used to express the oblique derivative in the boundary condition which has to be met by the solution. The relation to the classical formulation was also shown. The main result concerns the coerciveness (ellipticity) of a bilinear form associated with the problem under consideration. The Lax-Milgram theorem was used to decide about the existence, uniqueness and stability of the weak solution of the problem. Finally, a clear geometrical interpretation was found for a constant in the coerciveness inequality, and the convergence of approximation solutions constructed by means of the Galerkin method was proved. [ABSTRACT FROM AUTHOR]

We consider the Kawahara equation, a fifth order Korteweg–de Vries type equation, posed on a bounded interval. The first result of the article is related to the well-posedness in weighted Sobolev spaces, which one was shown using a general version of the Lax–Milgram Theorem. With respect to the control problems, we will prove two results. First, if the control region is a neighborhood of the right endpoint, an exact controllability result in weighted Sobolev spaces is established. Lastly, we show that the Kawahara equation is controllable by regions on L 2 Sobolev space, the so-called regional controllability , that is, the state function is exact controlled on the left part of the complement of the control region and null controlled on the right part of the complement of the control region. [ABSTRACT FROM AUTHOR]

Abstract: In this letter we introduce the concept of a weak solution for a damped linear equation with Dirichlet boundary conditions and impulses. We use the classical Lax–Milgram Theorem to reveal the variational structure of the problem and get the existence and uniqueness of weak solutions as critical points. This will allow us in the future to deal with the corresponding nonlinear problems and look for solutions as critical points of weakly lower semicontinuous functionals. [Copyright &y& Elsevier]