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Jarkko Siltakoski

Publications and source records attributed to Jarkko Siltakoski.

13 recordsLinked to original sources
arXiv2026

Higher integrability for parabolic double phase equations with an improved gap bound

We prove a local higher integrability result for the gradient of Hölder continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } Ω_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class $\mathcal{Z}^κ(Ω_T)$ for some $κ\in (0,\infty)$. The functions in this class satisfy a one-sided pointwise bound that controls how fast $a$ can grow away from its zero set, and the class contains the Hölder continuous functions. We also impose a mild almost increasing condition on $a$, which motivates the introduction of a new mollification, which we call the slanted Steklov average. For $u \in C^{0,γ,γ/q}_{\mathrm{loc}}(Ω_T)$ with $γ\in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q \le p + \frac{qκ}{q - 2γ}. \end{equation} The new gap condition \eqref{eq:G} is purely parabolic in nature and is stricter than the optimal gap relation associated with the Lavrentiev phenomenon for the elliptic double phase functional.

math.AP↗
arXiv2026

Existence of variational solutions to doubly nonlinear systems in nondecreasing domains

For $q \in (0, \infty)$, we consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_ξf(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. We assume that $x \mapsto f(x,u,ξ)$ is integrable, that $(u,ξ) \mapsto f(x,u,ξ)$ is convex, and that $f$ satisfies a $p$-coercivity condition for some $p \in (1,\infty)$. However, we do not impose any specific growth condition from above on $f$. For nondecreasing domains that merely satisfy $\mathcal{L}^{n+1}(\partial E) = 0$, we prove the existence of variational solutions $u \in C^{0}([0,T];L^{q+1}(E,\mathbb{R}^N))$ via a nonlinear version of the method of minimizing movements. Moreover, under additional assumptions on $E$ and a $p$-growth condition on $f$, we show that $|u|^{q-1}u$ admits a weak time derivative in the dual $(V^{p,0}(E))^{\prime}$ of the subspace $V^{p,0}(E) \subset L^p(0,T;W^{1,p}(Ω,\mathbb{R}^N))$ that encodes zero boundary values.

math.AP↗
arXiv2025

Lipschitz regularity for parabolic double phase equations with gradient nonlinearity

We establish the local Lipschitz regularity in space for the viscosity solutions to the parabolic double phase equation of the form \[ \smash{\partial_{t}u-\operatorname{div} \left(|Du|^{p-2}D u+a(z)|D u|^{q-2}D u\right)=f(z, Du)} \] by employing the Ishii-Lions method. In addition, we obtain Hölder estimate in time which turns out to be sharp in the degenerate regime. Here, $1< p\leq q<\infty,$ and the coefficient $a\geq 0$ is assumed to be bounded, locally Lipschitz continuous in space, and continuous in time. Furthermore, the non-homogeneity $f$ is assumed to be continuous on $Ω\times \mathbb{R}\times \mathbb{R}^N,$ and to satisfy a suitable gradient growth condition. We also establish the equivalence between bounded viscosity solutions and weak solutions, under appropriate additional regularity assumption on the coefficient $a.$

math.AP↗
arXiv2025

Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains

We consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_ξf(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} with $q \in (0, \infty)$ in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. Further, we suppose that $x \mapsto f(x,u,ξ)$ is integrable, that $(u,ξ) \mapsto f(x,u,ξ)$ is convex, and that $f$ satisfies a $p$-growth and -coercivity condition for some $p>\max \big\{ 1,\frac{n(q+1)}{n+q+1} \big\}$. Merely assuming that $\mathcal{L}^{n+1}(\partial E) = 0$, we prove the existence of variational solutions $u \in L^\infty\big( 0,T;L^{q+1}(E,\mathbb{R}^N)\big)$. If $E$ does not shrink too fast, we show that for the solution $u$ constructed in the first step, $\vert u \vert^{q-1}u$ admits a distributional time derivative. Moreover, under suitable conditions on $E$ and the stricter lower bound $p \geq \frac{(n+1)(q+1)}{n+q+1}$, $u$ is continuous with respect to time.

math.AP↗
arXiv2022

Elliptic Harnack's inequality for a singular nonlinear parabolic equation in non-divergence form

We prove an elliptic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that has been proposed in stochastic game theory. This version of the inequality doesn't require the intrinsic waiting time and we get the estimate with the same time level on both sides of the inequality.

math.AP↗
arXiv2022

The bounded slope condition for parabolic equations with time-dependent integrands

In this paper, we study the Cauchy-Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \mbox{$\partial_t u - \operatorname{div} \left( D_ξf(t, Du)\right) = 0$ } & \mbox{in $Ω_T$}, \\[5pt] \mbox{$u = u_o$} & \mbox{on $\partial_{\mathcal{P}} Ω_T$},\\[5pt] \end{array} \right. \end{equation*} where $Ω\subset \mathbb{R}^n$ is a convex domain, $f:[0,T]\times\mathbb{R}^n \rightarrow \mathbb{R}$ is $L^1$-integrable in time and convex in the second variable. Assuming that the initial and boundary datum $u_o:\overlineΩ\rightarrow \mathbb{R}$ satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.

math.AP↗
arXiv2020

Recovering a variable exponent

We consider an inverse problem of recovering the non-linearity in the one dimensional variable exponent $p(x)$-Laplace equation from the Dirichlet-to-Neumann map. The variable exponent can be recovered up to the natural obstruction of rearrangements. The main technique is using a Müntz-Szász theorem after reducing the problem to determining a function from its $L^p$-norms.

math.AP↗
arXiv2019

Equivalence between radial solutions of different non-homogeneous $p$-Laplacian type equations

We study radial viscosity solutions to the equation \[ -\ |Du\ |^{q-2}Δ_{p}^{N}u=f(\ |x\ |)\quad\text{in }B_{R}\subset\mathbb{R}^{N}, \] where $f\in C[0,R)$, $p,q\in(1,\infty)$ and $N\geq2$. Our main result is that $u(x)=v(\ |x\ |)$ is a bounded viscosity supersolution if and only if $v$ is a bounded weak supersolution to $-κΔ_{q}^{d}v=f$ in $(0,R)$, where $κ>0$ and $Δ_{q}^{d}$ is heuristically speaking the radial $q$-Laplacian in a fictitious dimension $d$. As a corollary we obtain the uniqueness of radial viscosity solutions. However, the full uniqueness of solutions remains an open problem.

math.AP↗
arXiv2019

Equivalence of viscosity and weak solutions for a $p$-parabolic equation

We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-Δ_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $p\geq2$.

math.AP↗
arXiv2017

Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian

We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,α}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Radó-type removability theorem.

math.AP↗