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where
$\alpha>n$
and
$\beta \in (0,1)$
. In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term
$R_{\alpha ,\beta }(\varphi )$
and inhomogeneous term
$G_{\alpha ,\beta }(g)$
, respectively. Second, we obtain some space–time estimates for
$G_{\alpha ,\beta }(g).$
Based on these estimates, we prove that the continuity of
$R_{\alpha ,\beta }(\varphi )(t,x)$
and the Hölder continuity of
$G_{\alpha ,\beta }(g)(t,x)$
on
$\mathbb {R}^{1+n}_+,$
which implies a Moser–Trudinger-type estimate for
$G_{\alpha ,\beta }.$
Then, for a newly introduced
$L^{q}_{t}L^p_{x}$
-capacity related to the space–time fractional dissipative operator
$\partial ^{\beta }_{t}+(-\Delta )^{\alpha /2},$
we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in
$\mathbb {R}^{1+n}_+$
by using the Strichartz estimates and the Moser–Trudinger-type estimate for
$G_{\alpha ,\beta }.$
A strong-type estimate of the
$L^{q}_{t}L^p_{x}$
-capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the
$L^{q}_{t}L^p_{x}$
-capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).
We prove that any continuous function can be locally approximated at a fixed point
$x_{0}$
by an uncountable family resistant to disruptions by the family of continuous functions for which
$x_{0}$
is a fixed point. In that context, we also consider the property of quasicontinuity.
Hardin and Taylor proved that any function on the reals—even a nowhere continuous one—can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman, who provided upper and lower frontiers (in the subgroup lattice of $\mathrm{Homeo}^+(\mathbb {R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder’s Theorem (that every Archimedean group is abelian).
Savin [‘
$\mathcal {C}^{1}$
regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal.3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.
Given $\beta \in (1,2]$, let $T_{\beta }$ be the $\beta $-transformation on the unit circle $[0,1)$ such that $T_{\beta }(x)=\beta x\pmod 1$. For each $t\in [0,1)$, let $K_{\beta }(t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^{n}_{\beta }(x): n\ge 0\}$ never hits the open interval $(0,t)$. Kalle et al [Ergod. Th. & Dynam. Sys.40(9) (2020) 2482–2514] proved that the Hausdorff dimension function $t\mapsto \dim _{H} K_{\beta }(t)$ is a non-increasing Devil’s staircase. So there exists a critical value $\tau (\beta )$ such that $\dim _{H} K_{\beta }(t)>0$ if and only if $t<\tau (\beta )$. In this paper, we determine the critical value $\tau (\beta )$ for all $\beta \in (1,2]$, answering a question of Kalle et al (2020). For example, we find that for the Komornik–Loreti constant$\beta \approx 1.78723$, we have $\tau (\beta )=(2-\beta )/(\beta -1)$. Furthermore, we show that (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps, with $\tau (1+)=0$ and $\tau (2)=1/2$; and (iii) there exists an open set $O\subset (1,2]$, whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, convex, and strictly decreasing on each connected component of O. Consequently, the dimension $\dim _{H} K_{\beta }(t)$ is not jointly continuous in $\beta $ and t. Our strategy to find the critical value $\tau (\beta )$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.
The cusped hyperbolic n-orbifolds of minimal volume are well known for
$n\leq 9$
. Their fundamental groups are related to the Coxeter n-simplex groups
$\Gamma _{n}$
. In this work, we prove that
$\Gamma _{n}$
has minimal growth rate among all non-cocompact Coxeter groups of finite covolume in
$\textrm{Isom}\mathbb H^{n}$
. In this way, we extend previous results of Floyd for
$n=2$
and of Kellerhals for
$n=3$
, respectively. Our proof is a generalization of the methods developed together with Kellerhals for the cocompact case.
According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.
The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension
$k\leq n$
in sub-Riemannian Heisenberg groups
${\mathbb H}^{n}$
. For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between
${\mathbb H}$
-rectifiability and ‘Lipschitz’
${\mathbb H}$
-rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.
We give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions
$f_\alpha:[0,1]\to{\mathbb R}$
parameterised by
$\alpha\in(-2,2)$
. We show that
$f_\alpha$
has a unique maximiser in
$[0,1/2]$
if and only if there does not exist a Littlewood polynomial that has
$\alpha$
as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with
$\alpha\in(-2,1/2]\cup(1,2)$
. For
$(1/2,1]$
, we show that the step roots are dense in that interval. If
$\alpha\in (1/2,1]$
is a step root, then the set of maximisers of
$f_\alpha$
is an explicitly given perfect set with Hausdorff dimension
$1/(n+1)$
, where n is the degree of the minimal Littlewood polynomial that has
$\alpha$
as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to
$[-2,-1/2]\cup[1/2,2]$
.
In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.
In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.
By developing a Green's function representation for the solution of the boundary value problem we study existence, uniqueness, and qualitative properties (e.g., positivity or monotonicity) of solutions to these problems. We apply our methods to fractional order differential equations. We also demonstrate an application of our methodology both to convolution equations with nonlocal boundary conditions as well as those with a nonlocal term in the convolution equation itself.
We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every
$C^1$
curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.
We show that a computable function
$f:\mathbb R\rightarrow \mathbb R$
has Luzin’s property (N) if and only if it reflects
$\Pi ^1_1$
-randomness, if and only if it reflects
$\Delta ^1_1({\mathcal {O}})$
-randomness, and if and only if it reflects
${\mathcal {O}}$
-Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever
$f(x)$
is R-random, then x is R-random as well. If additionally f is known to have bounded variation, then we show f has Luzin’s (N) if and only if it reflects weak-2-randomness, and if and only if it reflects
$\emptyset '$
-Kurtz randomness. This links classical real analysis with algorithmic randomness.
Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant
$1$
, the identity function
${\mathbf {x}}$
, and such that whenever f and g are in the set,
$f+g,fg$
and
$f^g$
are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below
$2^{2^{\mathbf {x}}}$
. Here we prove that the set of asymptotic classes within any Archimedean class of Skolem functions has order type
$\omega $
. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below
$2^{n^{\mathbf {x}}}$
. We deduce an epsilon-zero upper bound for the fragment below
$2^{{\mathbf {x}}^{\mathbf {x}}}$
, improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations.
We prove the Decomposability Conjecture for functions of Baire class
$2$
from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.
We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not
$\sigma $
-continuous with closed witnesses.
We present some inequalities for the mappings defined by Dragomir [‘Two mappings in connection to Hadamard’s inequalities’, J. Math. Anal. Appl.167 (1992), 49–56]. We analyse known inequalities connected with these mappings using a recently developed method connected with stochastic orderings and Stieltjes integrals. We show that some of these results are optimal and others may be substantially improved.
In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.