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The aim of this paper is to provide and numerically test in the presence of measurement noise a procedure for target classification in wave imaging based on comparing frequency-dependent distribution descriptors with precomputed ones in a dictionary of learned distributions. Distribution descriptors for inhomogeneous objects are obtained from the scattering coefficients. First, we extract the scattering coefficients of the (inhomogeneous) target from the perturbation of the reflected waves. Then, for a collection of inhomogeneous targets, we build a frequency-dependent dictionary of distribution descriptors and use a matching algorithm in order to identify a target from the dictionary up to some translation, rotation and scaling.
We study a free boundary problem of the form: ut = uxx + f(t, u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) – α(t) and g′(t) = −ux(t, g(t)) + β(t), where β(t) and α(t) are positive T-periodic functions, f(t, u) is a Fisher–KPP type of nonlinearity and T-periodic in t. This problem can be used to describe the spreading of a biological or chemical species in time-periodic environment, where free boundaries represent the spreading fronts of the species. We study the asymptotic behaviour of bounded solutions. There are two T-periodic functions α0(t) and α*(t; β) with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β< α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, that is, h(t) – g(t) → +∞ and u(t, ⋅ + ct) → 1 with
$c\in (-\overline{l},\overline{r})$
, where
$ \overline{l}:=\frac{1}{T}\int_{0}^{T}l(s)ds$
,
$\overline{r}:=\frac{1}{T}\int_{0}^{T}r(s)ds$
, the T-periodic functions −l(t) and r(t) are the asymptotic spreading speeds of g(t) and h(t) respectively (furthermore, r(t) > 0 > −l(t) when 0 < β < α < α0; r(t) = 0 > −l(t) when 0 < β < α = α0;
$0 \gt \overline{r} \gt -\overline{l}$
when 0 < β < α0 < α < α*); (i-2) vanishing, that is,
$\lim\limits_{t \to \mathcal {T}}h(t) = \lim\limits_{t \to \mathcal {T}}g(t)$
and
$\lim\limits_{t \to \mathcal {T}}\max\limits_{g(t)\leq x\leq h(t)} u(t,x)=0$
, where
$\mathcal {T}$
is some positive constant; (i-3) transition, that is, g(t) → −∞, h(t) → −∞,
$0<\lim\limits_{t \to \infty}[h(t)-g(t)] \lt +\infty$
and u(t, ⋅) → V(t, ⋅), where V is a T-periodic solution with compact support. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.
We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.
We study the existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Results on the existence of almost periodic-type solutions (including, periodic, almost periodic, asymptotically almost periodic and almost automorphic solutions) are proved. Some examples of partial differential equations with state-dependent delay arising in population dynamics are presented.
We analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.
In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.
We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.
The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.
We prove that the fractional Yamabe equation
${\rm {\cal L}}_\gamma u = \vert u \vert ^{((4\gamma )/(Q-2\gamma ))}u$
on the Heisenberg group ℍn has [n + 1/2] sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where
${\rm {\cal L}}_\gamma $
denotes the CR fractional sub-Laplacian operator on ℍn, Q = 2n + 2 is the homogeneous dimension of ℍn, and
$\gamma \in \bigcup\nolimits_{k = 1}^n [k,((kQ)/Q-1)))$
. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere S2n + 1 combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group U(n + 1).
We present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical case s = 1 in [23, 24] respectively.
where the function a(x, t, ξ) satisfies (p, q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps
$x\mapsto a(x,t,\xi )$
under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives
$D_xa(\cdot ,\cdot ,\xi )$
are contained in the class
$L^\alpha (0,T;L^\beta (\Omega ))$
, where the integrability exponents
$\alpha ,\beta $
are coupled by
for some κ ∈ (0,1). For the gap between the two growth exponents we assume
$$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$
Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative
$u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$
. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.
This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type
$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$
where ε > 0 is a parameter, s, t ∈ (0, 1) are such that 2s+2t>3, A:ℝ3 → ℝ3 is a smooth magnetic potential, (−Δ)As is the fractional magnetic Laplacian, V:ℝ3 → ℝ is a continuous electric potential and f:ℝ → ℝ is a C1 subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for e > 0 small enough.
where
$\tau :{\open R}^n\to {\open R}^n$
is a general function. In particular, for the linear choices
$\tau (x)=0$
,
$\tau (x)=x$
and
$\tau (x)={x}/{2}$
this covers the well-known Kohn–Nirenberg, anti-Kohn–Nirenberg and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions τ and here we investigate the corresponding calculus in the model case of
${\open R}^n$
. We also give examples of nonlinear τ appearing on the polarized and non-polarized Heisenberg groups.
We investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball
$B_1(0)\subset \mathbb{R}^n$
Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.
We give a unified approach to strong maximum principles for a large class of nonlocal operators of order s ∈ (0, 1) that includes the Dirichlet, the Neumann Restricted (or Regional) and the Neumann Semirestricted Laplacians.
We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.
In this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.
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.
In this paper, we study a class of Brezis–Nirenberg problems for nonlocal systems, involving the fractional Laplacian
$(-\unicode[STIX]{x1D6E5})^{s}$
operator, for
$0<s<1$
, posed on settings in which Sobolev trace embedding is noncompact. We prove the existence of infinitely many solutions in large dimension, namely when
$N>6s$
, by employing critical point theory and concentration estimates.
We establish existence of weighted Hardy and Rellich inequalities on the spaces
$L_{p}(\unicode[STIX]{x1D6FA})$
, where
$\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$
with
$K$
a closed convex subset of
$\mathbf{R}^{d}$
. Let
$\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
denote the boundary of
$\unicode[STIX]{x1D6FA}$
and
$d_{\unicode[STIX]{x1D6E4}}$
the Euclidean distance to
$\unicode[STIX]{x1D6E4}$
. We consider weighting functions
$c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$
with
$c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$
and
$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$
. Then the Hardy inequalities take the form
with
$H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$
. The constants
$b_{p},d_{p}$
depend on the weighting parameters
$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$
and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.
Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.
We consider a large class of
$1+1$
-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf–Cole solutions to the KPZ equation.
We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler–Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether’s theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler–Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.