Analysis of PDEs
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Showing new listings for Thursday, 14 November 2024
- [1] arXiv:2411.08175 [pdf, html, other]
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Title: Well-posedness of a Variable-Exponent Telegraph Equation Applied to Image DespecklingComments: 33 pages, 19 figures, 3 tablesSubjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
In this paper, we present a telegraph diffusion model with variable exponents for image despeckling. Moving beyond the traditional assumption of a constant exponent in the telegraph diffusion framework, we explore three distinct variable exponents for edge detection. All of these depend on the gray level of the image or its gradient. We rigorously prove the existence and uniqueness of weak solutions of our model in a functional setting and perform numerical experiments to assess how well it can despeckle noisy gray-level images. We consider both a range of natural images contaminated by varying degrees of artificial speckle noise and synthetic aperture radar (SAR) images. We finally compare our method with the nonlocal speckle removal technique and find that our model outperforms the latter at speckle elimination and edge preservation.
- [2] arXiv:2411.08362 [pdf, html, other]
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Title: Large-scale boundary estimates of parabolic homogenization over rough boundariesSubjects: Analysis of PDEs (math.AP)
In this paper, for a family of second-order parabolic system or equation with rapidly oscillating and time-dependent periodic coefficients over rough boundaries, we obtain the large-scale boundary estimates, by a quantitative approach. The quantitative approach relies on approximating twice: we first approximate the original parabolic problem over rough boundary by the same equation over a non-oscillating boundary and then approximate the oscillating equation over a non-oscillating boundary by its homogenized equation over the same non-oscillating boundary.
- [3] arXiv:2411.08366 [pdf, other]
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Title: Stability of the catenoid for the hyperbolic vanishing mean curvature equation in 4 spatial dimensionsComments: 105 pages, 1 figureSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
We establish the asymptotic stability of the catenoid, as a nonflat stationary solution to the hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space $\mathbb{R}^{1 + (n + 1)}$ for $n = 4$. Our main result is under a ``codimension-$1$'' assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \geq 5$ case addressed by Lührmann-Oh-Shahshahani arXiv:2212.05620, proving catenoid stability in $4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties in the $n = 3$ case, the strong Huygens principle, as well as a miraculous cancellation in the source term, plays an important role in arXiv:2409.05968 to obtain strong late time tails. In $n = 4$ dimensions, without these special structural advantages, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy of estimates with higher $r^p$-weights so that an improved pointwise decay can be established. We expect this to be applicable for proving improved late time tails of other quasilinear wave equations in even dimensions or wave equations with inverse square potential.
- [4] arXiv:2411.08428 [pdf, html, other]
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Title: Partially concentrating solutions for systems with Lotka-Volterra type interactionsSubjects: Analysis of PDEs (math.AP)
In this paper we consider the existence of standing waves for a coupled system of $k$ equations with Lotka-Volterra type interaction. We prove the existence of a standing wave solution with all nontrivial components satisfying a prescribed asymptotic profile. In particular, the $k-1$-last components of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. We analyze first in detail the result with three equations since this is the first case in which the coupling has a role contrary to what happens when only two densities appear. We also discuss the existence of solutions of this form for systems with other kind of couplings making a comparison with Lotka-Volterra type systems.
- [5] arXiv:2411.08585 [pdf, html, other]
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Title: Hardy type inequalities with mixed cylindrical-spherical weights: the general caseComments: 30 pagesSubjects: Analysis of PDEs (math.AP)
We continue our investigation of Hardy-type inequalities involving combinations of cylindrical and spherical weights. Compared to [Cora-Musina-Nazarov, Ann. Sc. Norm. Sup., 2024], where the quasi-spherical case was considered, we handle the full range of allowed parameters. This has led to the observation of new phenomena related to lack of compactness.
- [6] arXiv:2411.08614 [pdf, html, other]
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Title: Longtime and chaotic dynamics in microscopic systems with singular interactionsSubjects: Analysis of PDEs (math.AP)
This paper investigates the long time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of $N$ interacting point particles, where the time evolution of the joint distribution $f_N(t)$ is governed by the Liouville equation. Our primary objective is to analyze the system's behavior over extended time intervals, focusing on stability, potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime where $N \gg 1$, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases it is possible to derive uniform bounds in $ L^2 $, both over time and with respect to the number of particles, on the marginals $ \left(f_{k,N}\right)_{1\leq k \leq N}$, irrespective of the initial state's chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels surpassing the traditional $L^d$ regularity barriers, $K \in W^{\frac{-2}{d+2},d+2}(\mathbb{T}^d)$, where $\mathbb{T}$ denotes the $1$-torus and $d\geq2$ is the dimension. Finally, we address the highly singular case of $K \in H^{-1}(\mathbb{T}^d)$ within high-temperature regimes, offering new insights into the behavior of such systems.
- [7] arXiv:2411.08623 [pdf, html, other]
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Title: Non-local homogenization limits of discrete elastic spring network models with random coefficientsSubjects: Analysis of PDEs (math.AP)
This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive - in an appropriate scaling regime - a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers.
- [8] arXiv:2411.08657 [pdf, html, other]
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Title: Well-posedness and inverse problems for the nonlocal third-order acoustic equation with time-dependent nonlinearitySubjects: Analysis of PDEs (math.AP)
In this paper, we study the inverse problems of simultaneously determining a potential term and a time-dependent nonlinearity for the nonlinear Moore-Gibson-Thompson equation with a fractional Laplacian. This nonlocal equation arises in the field of peridynamics describing the ultrasound waves of high amplitude in viscous thermally fluids. We first show the well-posedness for the considered nonlinear equations in general dimensions with small exterior data. Then, by applying the well-known linearization approach and the unique continuation property for the fractional Laplacian, the potential and nonlinear terms are uniquely determined by the Dirichlet-to-Neumann (DtN) map taking on arbitrary subsets of the exterior in the space-time domain. The uniqueness results for general nonlinearities to some extent extend the existing works for nonlocal wave equations. Particularly, we also show a uniqueness result of determining a time-dependent nonlinear coefficient for a 1-dimensional fractional Jordan-Moore-Gibson-Thompson equation of Westervelt type.
- [9] arXiv:2411.08713 [pdf, html, other]
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Title: An existence result in annular regions times conical shells and its application to nonlinear Poisson systemsComments: 18 pages, 2 figuresSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
We provide a new existence result for abstract nonlinear operator systems in normed spaces, by means of topological methods. The solution is located within the product of annular regions and conical shells. The theoretical result possesses a wide range of applicability, which, for concreteness, we illustrate in the context of systems of nonlinear Poisson equations subject to homogeneous Dirichlet boundary conditions. For the latter problem we obtain existence and localization of solutions having all components nontrivial. This is also illustrated with an explicit example in which we also furnish a numerically approximated solution, consistent with the theoretical results.
New submissions (showing 9 of 9 entries)
- [10] arXiv:2411.08080 (cross-list from math.NA) [pdf, html, other]
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Title: Jacobi convolution polynomial for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite intervalSubjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution polynomials as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution polynomial is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator.
- [11] arXiv:2411.08146 (cross-list from math.CA) [pdf, html, other]
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Title: Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$Comments: 11 pagesSubjects: Classical Analysis and ODEs (math.CA); Analysis of PDEs (math.AP); Spectral Theory (math.SP)
Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in $\mathbb{C}^2$. They are also spherical harmonics (i.e., Laplacian eigenfunctions) on $\mathbb{S}^3 \subset \mathbb{R}^4$. In this paper, we prove that these functions tend to be equidistributed on $\mathbb{S}^3$, based on an estimate of the auto-correlation of the Rudin-Shapiro sequences. Moreover, we identify the semiclassical measure associated to these spherical harmonics by the singular measure supported on the family of Clifford tori in $\mathbb{S}^3$. In particular, this demonstrates a new localization pattern in the study of Laplacian eigenfunctions.
- [12] arXiv:2411.08387 (cross-list from q-bio.PE) [pdf, html, other]
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Title: Steady-State and Dynamical Behavior of a PDE Model of Multilevel Selection with Pairwise Group-Level CompetitionComments: 60 pages. 12 figuresSubjects: Populations and Evolution (q-bio.PE); Analysis of PDEs (math.AP); Physics and Society (physics.soc-ph)
Evolutionary competition often occurs simultaneously at multiple levels of organization, in which traits or behaviors that are costly for an individual can provide collective benefits to groups to which the individual belongs. Building off of recent work that has used ideas from game theory to study evolutionary competition within and among groups, we study a PDE model for multilevel selection that considers group-level evolutionary dynamics through a pairwise conflict depending on the strategic composition of the competing groups. This model allows for incorporation of group-level frequency dependence, facilitating the exploration for how the form of probabilities for victory in a group-level conflict can impact the long-time support for cooperation via multilevel selection. We characterize well-posedness properties for measure-valued solutions of our PDE model and apply these properties to show that the population will converge to a delta-function at the all-defector equilibrium when between-group selection is sufficiently weak. We further provide necessary conditions for the existence of bounded steady state densities for the multilevel dynamics of Prisoners' Dilemma and Hawk-Dove scenarios, using a mix of analytical and numerical techniques to characterize the relative strength of between-group selection required to ensure the long-time survival of cooperation via multilevel selection. We also see that the average payoff at steady state appears to be limited by the average payoff of the all-cooperator group, even for games in which groups achieve maximal average payoff at intermediate levels of cooperation, generalizing behavior that has previously been observed in PDE models of multilevel selection with frequency-indepdent group-level competition.
- [13] arXiv:2411.08787 (cross-list from nlin.SI) [pdf, html, other]
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Title: Stability analysis of breathers for coupled nonlinear Schrodinger equationsComments: 59 pagesSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Pattern Formation and Solitons (nlin.PS)
We investigate the spectral stability of non-degenerate vector soliton solutions and the nonlinear stability of breather solutions for the coupled nonlinear Schrodinger (CNLS) equations. The non-degenerate vector solitons are spectrally stable despite the linearized operator admits either embedded or isolated eigenvalues of negative Krein signature. The nonlinear stability of breathers is obtained by the Lyapunov method with the help of the squared eigenfunctions due to integrability of the CNLS equations.
- [14] arXiv:2411.08829 (cross-list from math.FA) [pdf, html, other]
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Title: Embeddings of anisotropic Sobolev spaces into spaces of anisotropic H\"{o}lder-continuous functionsJournal-ref: Demonstr. Math. 57:1 (2024), art. 20240079, 10 ppSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP)
We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.
Cross submissions (showing 5 of 5 entries)
- [15] arXiv:1804.06303 (replaced) [pdf, other]
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Title: Symmetry operators and generation of symmetry transformations of partial differential equationsComments: Enhanced and updated version; 23 pages; see also arXiv:0803.3688Journal-ref: Nausivios Chora Vol. 7 (2018) pp. C31-C48 ; https://nausivios.hna.gr/docs/2018C3.pdfSubjects: Analysis of PDEs (math.AP)
The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations, they present certain conceptual difficulties in the case of matrix-valued PDEs; for example, the usual differential-operator representation of the symmetry-generating vector fields is not possible in this case. In this article an algebraic approach to the symmetry problem of PDEs - both scalar and matrix-valued - is described, based on abstract operators (characteristic derivatives) that admit a standard differential-operator representation in the case of scalar-valued PDEs. A number of examples are given.
- [16] arXiv:2112.02871 (replaced) [pdf, other]
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Title: Variational inequality solutions and finite stopping time for a class of shear-thinning flowsJournal-ref: Annali di Matematica Pura ed Applicata, 2024, 203 (6), pp.2591-2612Subjects: Analysis of PDEs (math.AP)
The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald-DeWaele law) in dimension $N \in \{2,3\}$. We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald-DeWaele, Carreau-Yasuda, Herschel-Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a $p$-Laplacian for the symmetrized gradient for $p \in [1,2)$.
- [17] arXiv:2208.01842 (replaced) [pdf, html, other]
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Title: Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesicsComments: This article replaces articles arXiv:2208.01842 and arXiv:2205.05860, arXiv:2205.05860 is withdrawnSubjects: Analysis of PDEs (math.AP)
We consider a Lorentzian metric in $\mathbb{R}\times\mathbb{R}^n$. We show that if we know the lengths of the space-time geodesics starting at $(0,y,\eta)$ when $t=0$, then we can recover the metric at $y$. We prove the rigidity of Lorentzian metrics. We also prove a variant of the rigidity property for the case of null-geodesics: if two metrics are close and if corresponding null-geodesics have equal Euclidian lengths then the metrics are equal.
- [18] arXiv:2303.10485 (replaced) [pdf, html, other]
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Title: The soliton resolution conjecture for the Boussinesq equationComments: 43 pages, 11 figuresSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We analyze the Boussinesq equation on the line with Schwartz initial data belonging to the physically relevant class of global solutions. In a recent paper, we determined ten main asymptotic sectors describing the large $(x,t)$-behavior of the solution, and for each of these sectors we provided the leading order asymptotics in the case when no solitons are present. In this paper, we give a formula valid in the asymptotic sector $x/t \in (1,M]$, where $M$ is a large positive constant, in the case when solitons are present. Combined with earlier results, this validates the soliton resolution conjecture for the Boussinesq equation everywhere in the $(x,t)$-plane except in a number of small transition zones.
- [19] arXiv:2304.12430 (replaced) [pdf, html, other]
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Title: Weak solutions for weak turbulence models in electrostatic plasmasSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Plasma Physics (physics.plasm-ph)
The weak turbulence model, also known as the quasilinear theory in plasma physics, has been a cornerstone in modeling resonant particle-wave interactions in plasmas. This reduced model stems from the Vlasov-Poisson/Maxwell system under the weak turbulence assumption, incorporating the random phase approximation and ergodicity. The interaction between particles and waves (plasmons) can be treated as a stochastic process, whose transition probability bridges the momentum space and the spectral space. Therefore, the operators on the right hand side resemble collision forms, such as those in Boltzmann and Landau interacting models. For them, there have been results on well-posedness and regularity of solutions. However, as far as we know, there is no such preceding work for the quasilinear theory addressed in this manuscript.
In this paper, we establish the existence of global weak solutions for the system modeling electrostatic plasmas in one dimension. Our key contribution consists of associating the original integral-differential system to a degenerate inhomogeneous porous medium equation(PME) with nonlinear source terms, and leveraging advanced techniques from the PME literature. This approach opens a novel pathway for analyzing weak turbulence models in plasma physics. Moreover, our work offers new tools for tackling related problems in the broader context of nonlinear nonlocal PDEs. - [20] arXiv:2308.10221 (replaced) [pdf, html, other]
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Title: Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equationComments: 34 pages, introduction rewritten and Remark 1.6 addedSubjects: Analysis of PDEs (math.AP)
Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\eta>0$, $\beta>\frac{m\rho_1}{n-2-nm}$, $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$, $\beta_0>0$ and $\alpha_0=2\beta_0+1$. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution $f=f^{(m)}$ of the elliptic equation $\Delta (f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha/\beta}f(x)=\eta$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $\Delta\log g+\alpha_0 g+\beta_0x\cdot\nabla g=0$, $g>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha_0/\beta_0}g(x)=\eta$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=\Delta u^m$ and the logarithmic diffusion equation $u_t=\Delta\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to\infty$.
- [21] arXiv:2311.10618 (replaced) [pdf, html, other]
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Title: Metric viscosity solutions on metric spaces and on the Wasserstein spaceComments: 22 pagesSubjects: Analysis of PDEs (math.AP)
Viscosity solutions to the eikonal equation on a non-compact complete Riemannian manifold are crucial for understanding the underlying geometric and topological properties. The classical definition of viscosity solutions relies on the differential structure and does not directly extended to general metric spaces. In this work, we explore the (metric) viscosity solutions on metrics spaces, in particular, the Wasserstein space P_p(X) where X is a complete, separable, locally compact, non-compact geodesic space. We introduce a new and elegant definition of the metric viscosity solutions in this context and investigate their properties such as stability, uniqueness and comparison principle. Moreover, we study their relationship with two types of distance-like function and present two distinct methods to construct (strong) metric viscosity solutions on the Wasserstein space P_p(X).
- [22] arXiv:2401.16344 (replaced) [pdf, other]
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Title: Trace estimates for harmonic functions along circular arcs with applications to domain decomposition on overlapping disksSubjects: Analysis of PDEs (math.AP)
In this paper we derive several (and in many cases sharp) estimates for the $\mathrm{L}^2$-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and numerical range of the Dirchlet-to-Dirichlet (DtD) operator sending data on the boundary of the disk to the restriction of its harmonic extension along circular arcs inside the disk. The estimates we derive here have applications in the convergence analysis of the Schwarz domain decomposition method for overlapping disks in two dimensions. In particular, they allow us to establish a rigorous convergence proof for the discrete parallel Schwarz method applied to the Conductor-like Screening Model (COSMO) from theoretical chemistry in the two-disk case, and to derive error estimates with respect to the discretization parameter, the number of Schwarz iterations, and the geometry of the domain. Our analysis addresses challenges beyond classical domain decomposition theory, especially the weak enforcement of boundary conditions.
- [23] arXiv:2403.07672 (replaced) [pdf, html, other]
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Title: Quantitative estimates in almost periodic homogenization of parabolic systemsSubjects: Analysis of PDEs (math.AP)
We consider a family of second-order parabolic operators $\partial_t+\mathcal{L}_\varepsilon$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary Hölder and Lipschitz estimates as well as convergence rate. The estimates of fundamental solution and Green's function are also established. In contrast to periodic case, the main difficulty is that the corrector equation $(\partial_s+\mathcal{L}_1)(\chi^\beta_{j})=-\mathcal{L}_1(P^\beta_j) $ in $\mathbb{R}^{d+1}$ may not be solvable in the almost periodic setting for linear functions $P(y)$ and $\partial_t \chi_S$ may not in $B^2(\mathbb{R}^{d+1})$. Our results are new even in the case of time-independent coefficients.
- [24] arXiv:2406.01097 (replaced) [pdf, other]
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Title: A multiplicative inequality of Riesz transform type on general Riemannian manifoldsEl Maati Ouhabaz (IMB)Comments: Expanded version, three new section are addedSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
Given any complete Riemannian manifold $M$, we prove that for every $p \in (1, 2]$ and every $\epsilon > 0$, $$ \| \nabla f \|_p^2 \le C_\epsilon \| \Delta^{\frac{1}{2} + \epsilon} f \|_{p}\| \Delta^{\frac{1}{2} - \epsilon} f \|_{p}.$$The estimate is dimension free. This inequality is even proved in the abstract setting of generators of sub-Markov semigroups.
- [25] arXiv:2407.20954 (replaced) [pdf, html, other]
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Title: On the dimension of observable sets for the heat equationSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Complex Variables (math.CV)
We consider the heat equation on a bounded $C^1$ domain in $\mathbb{R}^n$ with Dirichlet boundary conditions. The primary aim of this paper is to prove that the heat equation is observable from any measurable set with a Hausdorff dimension strictly greater than $n - 1$. The proof relies on a novel spectral estimate for linear combinations of Laplace eigenfunctions, achieved through the propagation of smallness for solutions to Cauchy-Riemann systems as established by Malinnikova, and uses the Lebeau-Robbiano method. While this observability result is sharp regarding the Hausdorff dimension scale, our secondary goal is to construct families of sets with dimensions less than $n - 1$ from which the heat equation is still observable.
- [26] arXiv:2410.11659 (replaced) [pdf, html, other]
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Title: A quasilinear elliptic equation with absorption term and Hardy potentialComments: 47 pages,4 figuresSubjects: Analysis of PDEs (math.AP)
Here we study the positive solutions of the equation \begin{equation*} -\Delta _{p}u+\mu \frac{u^{p-1}}{\left\vert x\right\vert ^{p}}+\left\vert x\right\vert ^{\theta }u^{q}=0,\qquad x\in \mathbb{R}^{N}\backslash \left\{ 0\right\} \end{equation*}% where $\Delta _{p}u={div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ and $1<p<N,q>p-1,\mu ,\theta \in \mathbb{R}.$ We give a complete description of the existence and the asymptotic behaviour of the solutions near the singularity $0,$ or in an exterior domain. We show that the global solutions $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ are radial and give their expression according to the position of the Hardy coefficient $\mu $ with respect to the critical exponent $\mu _{0}=-(\frac{N-p}{p})^{p}.$ Our method consists into proving that any nonradial solution can be compared to a radial one, then making exhaustive radial study by phase-plane techniques. Our results are optimal, extending the known results when $\mu =0$ or $p=2$, with new simpler this http URL make in evidence interesting phenomena of nonuniqueness when $\theta +p=0$, and of existence of locally constant solutions when moreover $p>2$ .
- [27] arXiv:2405.03335 (replaced) [pdf, html, other]
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Title: Spectral properties of the resolvent difference for singularly perturbed operatorsSubjects: Spectral Theory (math.SP); Analysis of PDEs (math.AP)
We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators $\mathbf{A}+\mathbf{V}_1$ and $\mathbf{A}+\mathbf{V}_2$ in a domain $\Omega\subseteq \mathbb{R}^\mathbf{N}$ with perturbations $\mathbf{V}_1, \mathbf{V}_2$ generated by $V_1\mu,V_2\mu,$ where $\mu$ is a measure singular with respect to the Lebesgue measure and satisfying two-sided or one-sided conditions of Ahlfors type, while $V_1,V_2$ are weight functions subject to some integral conditions. As an important special case, spectral estimates for the difference of resolvents of two Robin realizations of the operator $\mathbf{A}$ with different weight functions are obtained. For the case when the support of the measure is a compact Lipschitz hypersurface in $\Omega$ or, more generally, a rectifiable set of Haußdorff dimension $d=\mathbf{N}-1$, the Weyl type asymptotics for eigenvalues is justified.