On the Dirichlet problem in cylindrical domains for evolution Ole\v{\i}nik--Radkevi\v{c} PDE's: a Tikhonov-type theorem

TL;DR

This paper establishes a Tikhonov-type theorem linking boundary regularity of solutions for certain evolution PDEs in cylindrical domains to the regularity of their spatial parts, with applications to degenerate Ornstein-Uhlenbeck operators.

Contribution

It proves a new boundary regularity criterion for evolution PDEs with hypoelliptic spatial operators, extending classical results to degenerate and hypoelliptic cases.

Findings

Boundary regularity in cylindrical domains is characterized by the spatial boundary regularity.

The theorem applies to PDEs with nonnegative characteristic form and hypoellipticity conditions.

A new criterion for boundary regularity of degenerate Ornstein-Uhlenbeck operators is derived.

Abstract

We consider the linear second order PDO's $$ \mathscr{L} = \mathscr{L}_0 - \partial_t : = \sum_{i,j =1}^N \partial_{x_i}(a_{i,j} \partial_{x_j} ) - \sum_{j=i}^N b_j \partial_{x_j} - \partial _t,$$and assume that $\mathscr{L}_0$ has nonnegative characteristic form and satisfies the Ole\v{\i}nik--Radkevi\v{c} rank hypoellipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for $\mathscr{L}$ and $\mathscr{L}_0$ on bounded open subsets of $\mathbb R^{N+1}$ and of $\mathbb R^{N}$, respectively. Our main result is the following Tikhonov-type theorem: Let $\mathcal{O}:= \Omega \times ]0, T[$ be a bounded cylindrical domain of $\mathbb R^{N+1}$, $\Omega \subset \mathbb R^{N},$ $x_0 \in \partial \Omega$ and $0 < t_0 < T.$ Then $z_0 = (x_0, t_0) \in \partial \mathcal{O}$ is $\mathscr{L}$-regular for $\mathcal{O}$ if and only if $x_0$ is $\mathscr{L}_0$-regular for $\Omega$. As an application, we derive a boundary regularity criterion for degenerate Ornstein--Uhlenbeck operators.

Equations

$$\mathcal{L}={\mathcal{L}}_0-{\partial }_t:=\sum _{i,j=1}^N{\partial }_{x_i}(a_{i,j}{\partial }_{x_j})-\sum _{j=i}^N{b}_j{\partial }_{x_j}-{\partial }_t,$$

$${\mathcal{L}}_0:=\sum _{i,j=1}^N{\partial }_{x_i}\left(a_{ij}{\partial }_{x_j}\right)+\sum _{j=1}^N{b}_j{\partial }_{x_j}$$

$$\mathcal{L}={\mathcal{L}}_0-{\partial }_t.$$

$$A(x):=(a_{ij}(x){)}_{i,j=1,\dots ,N}\ge 0\forall x\in X.$$

$$\underset{X}{\inf }{a}_{11}=:\alpha >0.$$

$$X_i=\sum _{j=1}^N{a}_{ij}{\partial }_{x_j},i=1,\dots ,N,\text{mbox}and{X}_0=\sum _{j=1}^N{b}_j{\partial }_{x_j}.$$

$$\{\begin{array}{l}{\mathcal{L}}_{0}u=0\text{mbox}in\Omega ,\\ u{|}_{\partial \Omega }=\phi ,\end{array}\text{mbox}and\{\begin{array}{l}({\mathcal{L}}_0-{\partial }_t)v=0\text{mbox}in\mathcal{O}:=\Omega \times ]0,T[,\\ v{|}_{\partial \mathcal{O}}=\psi ,\end{array}$$

$$H_{\phi }^{\Omega }\text{mbox}and{K}_{\psi }^{\mathcal{O}}.$$

$$\underset{x⟶x_0}{\lim }{H}_{\phi }^{\Omega }(x)=\phi (x_0)\forall \phi \in C(\partial \Omega ,\mathbb{R})$$

$$\left(\underset{(x,t)⟶(x_0,t_0)}{\lim }{K}_{\psi }^{\mathcal{O}}(x,t)=\psi (x_0,t_0)\forall \psi \in C(\partial \mathcal{O},\mathbb{R})\right).$$

$$\{x\in V|\underset{n\in \mathbb{N}}{\sup }{u}_n(x)<\infty \}$$

$$u:=\sup_{V}u_{n}\mbox{\quad is ${\mathcal{H}}$-harmonic in\ }V.$$

$$u(x)v(y)\ne u(y)v(x).$$

$$\underset{E\setminus V}{\inf }{S}_{x_0}>0$$

$${\overline{\mathcal{U}}}_{\phi }^V=\{u\in \overline{\mathcal{H}}(V)|\underset{x⟶\xi }{\mathrm{liminf}}u(x)\ge \phi (\xi )\forall \xi \in \partial V\}$$

$$H_{\phi }^V=:\inf {\overline{\mathcal{U}}}_{\phi }^V.$$

$$\{\begin{array}{l}u\in \mathcal{H}(V),\\ u{|}_{\partial V}=\phi .\end{array}$$

$$H_{\phi }^V=:\sup {\underline{\mathcal{U}}}_{\phi }^V,$$

$${\underline{\mathcal{U}}}_{\phi }^V=\{v\in \underline{\mathcal{H}}(V)|\underset{x⟶\xi }{\mathrm{limsup}}v(x)\le \phi (\xi )\forall \xi \in \partial V\}.$$

$$\underset{x⟶y}{\lim }{H}_{\phi }^V(x)=\phi (y)\forall \phi \in C(\partial V,\mathbb{R}).$$

$$\underset{x⟶y}{\mathrm{liminf}}u(x)\ge 0\forall y\in \partial V,$$

$$\mathcal{H}(V)=\{u\in C^{\infty }(V,\mathbb{R})|{\mathcal{L}}_{0}u=0\text{mbox}inV\}.$$

$$(E,{\mathcal{H}})\mbox{ is a {\it$\sigma^{\ast}\!\!$-harmonic space}}.$$

$$\mathcal{L}:=\sum _{i,j=1}^N{a}_{ij}{\partial }_{x_i{x}_j}+\sum _{j=1}^N{b}_j{\partial }_{x_j},$$

$$\underset{Y}{\inf }{a}_{11}:=\alpha >0\text{mbox}and\sum _{j=1}^N{a}_{jj}>0\text{mbox}inY.$$

$$h(x)=E(\lambda x_1)+(x_2^2+\cdots +x_N^2),x=(x_1,x_2,\dots ,x_N)\in {\mathbb{R}}^N,$$

$$E(s)=\exp (\varphi (s))-\exp (\varphi (0))$$

$$\varphi (s)=\sqrt{1+s^2},s\in \mathbb{R}.$$

$$\varphi (0)=1,\varphi (s)>1\forall s\ne 0,E(s)>0\forall s\ne 0,E(0)=0,$$

$${\varphi }^{\prime }(s)=\frac{s}{\sqrt{1+s^2}},{\varphi }^{\prime \prime }(s)=\frac{1}{(1+s^2{)}^{\frac{3}{2}}}.$$

Full text

[TABLE]

and assume that $\mathscr{L}_{0}$ has nonnegative characteristic form and satisfies the Oleǐnik–Radkevič rank hypoellipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for $\mathscr{L}$ and $\mathscr{L}_{0}$ on bounded open subsets of $\mathbb{R}^{N+1}$ and of $\mathbb{R}^{N}$, respectively.

Our main result is the following Tikhonov-type theorem:

Let $\mathcal{O}:=\Omega\times]0,T[$ be a bounded cylindrical domain of $\mathbb{R}^{N+1}$, $\Omega\subset\mathbb{R}^{N},$ $x_{0}\in\partial\Omega$ and $0<t_{0}<T$. Then $z_{0}=(x_{0},t_{0})\in\partial\mathcal{O}$ is $\mathscr{L}$-regular for $\mathcal{O}$ if and only if $x_{0}$ is $\mathscr{L}_{0}$-regular for $\Omega$.

As an application, we derive a boundary regularity criterion for degenerate Ornstein–Uhlenbeck operators.

1. Introduction

We consider linear second order partial differential operators of the type

[TABLE]

in an open set $X$ of ${\mathbb{R}}^{N}$, $N\geq 2,$ and their “evolution”counterpart in $X\times{\mathbb{R}}$

[TABLE]

We assume $\mathscr{L}_{0}$ in (1.1) is of non totally degenerate Oleǐnik and Radkevič type, i.e., we assume

• (H1)

$a_{ij}=a_{ji},b_{i}\in C^{\infty}(X,{\mathbb{R}})$ and

[TABLE]

Moreover

[TABLE]

• (H2)

$\mathrm{rank\ }\mathrm{Lie}\{X_{1},\ldots,X_{N},X_{0}\}(x)=N\qquad\forall x\in X,$ where,

[TABLE]

Hypotheses (H1) and (H2) imply that ${\mathscr{L}_{0}}$ is hypoelliptic in $X$ (see [OR73]), that is:

$\Omega$ open subset of $X$, $u\in\mathcal{D}^{\prime}(\Omega),{\mathscr{L}_{0}}u\in C^{\infty}(\Omega,{\mathbb{R}})\implies u\in C^{\infty}(\Omega,{\mathbb{R}}).$

The same assumptions (H1) and (H2) also imply that ${\mathscr{L}_{0}}-\partial_{t}$ is hypoelliptic in $X\times{\mathbb{R}}$.

We will show in Section 2 that ${\mathscr{L}_{0}}$ and ${\mathscr{L}_{0}}-\partial_{t}$ endow $X$ and $X\times{\mathbb{R}}$, respectively, with a local structure of $\sigma^{\ast}\!\!$-harmonic space, in the sense of [3], Chapter 6. As a consequence, in particular, the Dirichlet problems

[TABLE]

have a generalized solution in the sense of Perron–Wiener, for every bounded open set $\Omega\subset\subset X,$ for every $T>0$, and for every $\varphi\in C(\partial\Omega,{\mathbb{R}})$ and $\psi\in C(\partial\mathcal{O},{\mathbb{R}})$. We will denote such generalized solutions by, respectively,

[TABLE]

As usual, we say that a point $x_{0}\in\partial\Omega$ ($(x_{0},t_{0})\in\partial{\mathcal{O}}$) is ${\mathscr{L}_{0}}$-regular for $\Omega$ (${\mathscr{L}}$-regular for ${\mathcal{O}}$) if

[TABLE]

[TABLE]

The aim of this paper is to prove the following theorem:

Theorem 1.1**.**

Let $\Omega$ be a bounded open set with $\overline{\Omega}\subseteq X$, and let $x_{0}\in\partial\Omega$ and $t_{0}\in]0,T[$. Then, $x_{0}$ is ${\mathscr{L}_{0}}$-regular for $\Omega$ if and only if $(x_{0},t_{0})$ is ${\mathscr{L}_{0}}-\partial_{t}$-regular for ${\mathcal{O}}:=\Omega\times]0,T[$.

When ${\mathscr{L}}=\varDelta-\partial_{t}$ is the classical heat operator, our result re-establishes a theorem proved by Tikhonov in 1938 [Tik38]. Other proofs of the Tikhonov Theorem were given by Fulks in 1956 and in 1957 [Ful56, Ful57] and by Babuška and Výborný in 1962 [BV62]. Chan and Young extended the Tikhonov Theorem to parabolic operators with Hölder continuous coefficients in 1977 [CY77], and Arendt to parabolic operators with bounded measurable coefficients in 2000 [Are00]. The corresponding version for $p$-Laplacian-type evolution operators has been proved by Kilpeläinen and Lindqvist in 1996 [KL96] and by Banerjee and Garofalo in 2015 [BG15].

To the best of our knowledge, the only Tikhonov-type theorem for second order “evolution”sub-Riemannian PDO’s appearing in the literature is the result by Negrini [Neg83] in abstract $\beta$-harmonic spaces111For a definition of $\beta$-harmonic spaces see [CC72]..

This paper is organised as follows. In Section 2, all the notions and results from Potential Theory that we need are briefly recalled. In particular, we recall the notion of $\sigma^{\ast}\!\!$-harmonic space and then we prove that ${\mathscr{L}}_{0}$ and ${\mathscr{L}}$ endow $X$ and $X\times{\mathbb{R}}$, respectively, with a local structure of $\sigma^{\ast}\!\!$-harmonic space. In this way, we derive the existence of a generalized solution in the sense of Perron–Wiener in both our settings. Section 3 is devoted to two key results for the proof of the main theorem (Theorem 1.1), which is the content of Section 4. Finally, combining our Tikhonov-type theorem with a corollary of the Wiener–Landis-type criterion for Kolmogorov-type operators proved in [KLT18], we establish a geometric boundary regularity criterion for degenerate Ornstein–Uhlenbeck operators.

2. ${\mathscr{L}_{0}}$-harmonic and ${\mathscr{L}}$-harmonic spaces

2.1. The $\sigma^{\ast}\!\!$-harmonic space

For the readers’ convenience we recall the definition of $\sigma^{\ast}\!\!$-harmonic space supported on a an open set $E\subseteq{\mathbb{R}}^{p},p\geq 2$, and refer to Chapter 6 of the monograph [BLU07] for details.

Let ${\mathcal{H}}$ be a sheaf of functions in $E$ such that ${\mathcal{H}}(V)$ is a linear subspace of $C(V,{\mathbb{R}})$, for every open set $V\subseteq E$. The functions in ${\mathcal{H}}(V)$ are called ${\mathcal{H}}$-harmonic in $V.$ The open set $V$ is called ${\mathcal{H}}$-regular if

• $(i)$

$\overline{V}\subseteq E$ is compact;

• $(ii)$

for every $\varphi\in C(\partial V,{\mathbb{R}})$ there exists a unique function such that

$h_{\varphi}^{V}(x)\rightarrow\varphi(\xi)$ as $x\rightarrow\xi$, for every $\xi\in\partial V;$

• $(iii)$

$h_{\varphi}^{V}\geq 0$ if $\varphi\geq 0.$

A lower semicontinuous function $u:W\longrightarrow]-\infty,\infty],$ $W\subseteq E$ open, is called ${\mathcal{H}}$-superharmonic if

• $(i)$

$u\geq h_{\varphi}^{V}$ in $V$ for every ${\mathcal{H}}$-regular open set $V$ with $\overline{V}\subseteq W$ and for every $\varphi\in C(\partial V,{\mathbb{R}})$ with $\varphi\leq u|_{\partial V};$

• $(ii)$

$\{x\in W\ |\ u(x)<\infty\}$ is dense in $W.$

We denote by $\overline{{\mathcal{H}}}(W)$ the cone of the ${\mathcal{H}}$-superharmonic functions in $W.$

The couple $(E,{\mathcal{H}})$ is called a $\sigma^{\ast}\!\!$-harmonic space if the following axioms hold:

• (A1)

There exists a function $h\in{\mathcal{H}}(E)$ such that $\inf h>0$.

• (A2)

If $(u_{n})_{n\in\mathbb{N}}$ is a monotone increasing sequence of ${\mathcal{H}}$-harmonic functions in an open set $V\subseteq E$ such that

[TABLE]

is dense in $\Omega$, then

[TABLE]

• (A3)

The family of the ${\mathcal{H}}$-regular open sets is a basis of the Euclidean topology on $E$.

• (A4)

For every $x,y\in E$, $x\neq y$, there exist two nonnegative ${\mathcal{H}}$-superharmonic and continuous functions $u,v$ in $E$ such that

[TABLE]

• (A5)

For every $x_{0}\in E$ there exists a nonnegative ${\mathcal{H}}$-superharmonic and continuous function $S_{x_{0}}$ in $E$, such that $S_{x_{0}}(x_{0})=0$ and

[TABLE]

for every neighborhood $V$ of $x_{0}$.

We now recall some crucial results in $\sigma^{\ast}\!\!$-harmonic space theory; first of all the definition of Perron–Wiener solution to the Dirichlet problem.

Let $V$ be a bounded open set with $\overline{V}\subseteq E$, and let $\varphi:\partial V\longrightarrow{\mathbb{R}}$ be a bounded lower semicontinuous or upper semicontinuous function. Define

[TABLE]

and

[TABLE]

Then $H_{\varphi}^{V}$ is ${\mathcal{H}}$-harmonic in $\Omega.$ It is called the generalized Perron–Wiener solution to the Dirichlet problem

[TABLE]

We also have

[TABLE]

where,

[TABLE]

Here $\underline{{\mathcal{H}}}(V):=-\overline{\mathcal{H}}(V)$ denotes the cone of the ${\mathcal{H}}$-subharmonic functions in $V.$

A point $y\in\partial V$ is called ${\mathcal{H}}$-regular for $V$ if

[TABLE]

On the $\sigma^{\ast}\!\!$-harmonic space Bouligand Theorem holds. Indeed: a point $y\in\partial V$ is ${\mathcal{H}}$-regular for $V$ if and only if there exists a ${\mathcal{H}}$-barrier for $V$ at $y$, i.e., if there exists a function $b$ ${\mathcal{H}}$-superharmonic in $V\cap W,$ where $W$ is a neighborhood of $y,$ such that

• $(i)$

$b$ is ${\mathcal{H}}$-superharmonic;

• $(ii)$

$b(x)>0\ \forall x\in V\cap W$ and $b(x)\longrightarrow 0$ as $x\longrightarrow y.$

For our purposes it is important to recall that if $y\in\partial V$ is ${\mathcal{H}}$-regular for $V$ there exists a barrier function for $V$ at $y$ which is defined and ${\mathcal{H}}$-harmonic all over $V.$

Finally, we recall the minimum principle for ${\mathcal{H}}$-superharmonic functions.

Let $V$ be a bounded open set with $\overline{V}\subseteq E$ and let $u\in\overline{\mathcal{H}}(V).$ If

[TABLE]

then $u\geq 0$ in $V.$

2.2. The ${\mathscr{L}_{0}}$-harmonic space

Let $E$ be a bounded open subset of $X$ such that $\overline{E}\subseteq X.$ For every open set $V\subseteq E$ we let

[TABLE]

Then, $V\longmapsto{\mathcal{H}}(V)$ is a a sheaf of functions such that ${\mathcal{H}}(V)$ is a linear subspace of $C(V,{\mathbb{R}}).$

If $u\in{\mathcal{H}}(V)$ we will say that $u$ is ${\mathcal{H}}$-harmonic or ${\mathscr{L}}_{0}$-harmonic in $V.$

We have that

[TABLE]

Before showing this statement we remark that a $C^{2}$-function $u$ in a open set $V$ is ${\mathcal{H}}$-superharmonic if and only if ${\mathscr{L}_{0}}u\leq 0$ in $V$. This is a easy consequence of Picone’s maximum principle (see e.g. [KP16], page 547). Now we are ready to prove (2.3).

(A1) is satisfied since the constant functions are ${\mathscr{L}_{0}}$-harmonic.

(A2) -(A4) are proved in [KP16]. We would like to stress that our operators ${\mathscr{L}_{0}}$ are contained in the class considered in [KP16] since the rank condition (H2) implies that both ${\mathscr{L}_{0}}$ and ${\mathscr{L}_{0}}-\beta,$ for every $\beta\geq 0,$ are hypoelliptic.

The axiom (A5) follows from the following Lemma which seems to have an independent interest in its own right.

Lemma 2.1**.**

Let us consider a linear second order PDO of the kind

[TABLE]

where $a_{ij}=a_{ji},b_{j}$ are continuous functions in $\overline{Y}$, where $Y$ is a bounded open subset of ${{\mathbb{R}}^{{N}}}$. Suppose

[TABLE]

Then, for every $x_{0}\in Y$ there exists a function $h\in C^{\infty}(Y,{\mathbb{R}})$ such that

• $(i)$

$h(x_{0})=0$* and $h(x)>0$ for every $x\neq x_{0};$*

• $(ii)$

$\mathcal{L}h>0$* in $X.$*

Proof.

For the sake of simplicity we assume $x_{0}=0.$ We define

[TABLE]

where $\lambda>0$ will be fixed below. Moreover,

[TABLE]

and

[TABLE]

We have:

[TABLE]

[TABLE]

Hence

[TABLE]

On the other hand

[TABLE]

Therefore, letting

[TABLE]

we get

[TABLE]

If $\lambda$ is big enough, this implies

[TABLE]

Moreover

[TABLE]

The proof is complete. ∎

2.3. The ${\mathscr{L}}$-harmonic space

Let $\widehat{E}$ be a bounded open subset of $X\times{\mathbb{R}}$ such that $\overline{\widehat{E}}\subseteq X\times{\mathbb{R}}.$ For every open set $V\subseteq\widehat{E}$ we let

[TABLE]

Then, $V\longmapsto{\mathcal{K}}(V)$ is a a sheaf of functions making

[TABLE]

This can be proved just by proceeding as in subsection 2.2. We call ${\mathcal{K}}$-harmonic or ${\mathscr{L}}$-harmonic in a open set $V$ the solutions to ${\mathscr{L}}u=0$ in $V.$

Here we prove some typical results of the present ${\mathcal{K}}$-harmonic space, that we will need in the proof of the main theorem of this paper. We first show a “parabolic”minimum principle for ${\mathscr{L}}$-subharmonic functions in cylindrical domains.

Proposition 2.2**.**

Let $\Omega$ be a bounded open subset of $X$ such that $\overline{\Omega}\subseteq X$ and let $T>0$. Consider the cylindrical domain ${\mathcal{O}}:=\Omega\times]0,T[$ and define the “parabolic boundary”of ${\mathcal{O}}$ as follows

[TABLE]

Then, if $u\in\overline{{\mathcal{K}}}({\mathcal{O}})$ is such that

[TABLE]

we have $u\geq 0$ in ${\mathcal{O}}.$

Proof.

For every arbitrarily fixed $\widehat{T}\in]0,T[$ we let $\widehat{{\mathcal{O}}}=\Omega\times]0,\widehat{T}[.$ We will prove that $u\geq 0$ in $\widehat{{\mathcal{O}}}$. Since $\widehat{T}$ is arbitrarily fixed in $]0,T[$, this will give the proof of our lemma. To this end, given any $\varepsilon>0$, we define

[TABLE]

Since $u$ is ${\mathcal{K}}$-superharmonic in ${\mathcal{O}}$ and

[TABLE]

then $u_{\varepsilon}$ is ${\mathcal{K}}$-superharmonic in ${\mathcal{O}}$. Moreover

[TABLE]

and, for every $\xi\in\Omega,$

[TABLE]

By the minimum principle recalled in subsection 2.1, we have $u_{\varepsilon}\geq 0$ in $\widehat{{\mathcal{O}}}$. Letting $\varepsilon$ go to zero we have $u_{\varepsilon}\geq 0$ in $\widehat{{\mathcal{O}}}$, thus completing the proof. ∎

Proposition 2.3**.**

Let $\Omega\subseteq X$ be open and let $T_{0}$ and $T\in{\mathbb{R}},$ such that $0<T_{0}<T.$ Let ${\mathcal{O}}:=\Omega\times]0,T[$ and $u:{\mathcal{O}}\longrightarrow{\mathbb{R}}$ be such that the restrictions $u|_{\Omega\times]0,T_{0}[}$ and $u|_{\Omega\times]T_{0},T[}$ are ${\mathcal{K}}$-superharmonic. Then, if

[TABLE]

the function $u$ is ${\mathcal{K}}$-superharmonic in $\Omega\times]0,T[.$

Proof.

Since $u$ is lower semicontinuous in $\Omega\times]0,T_{0}[$ and in $\Omega\times]T_{0},T[$, the assumption (2.4) implies that u is lower semicontinuous in ${\mathcal{O}}=\Omega\times]0,T[.$

To prove that $u$ is ${\mathcal{K}}$-harmonic in ${\mathcal{O}}$ we will show the following claim.

*Claim. * For every $z\in{\mathcal{O}}$ there exists a basis $B_{z}$ of ${\mathcal{K}}$-regular neighborhoods of $V$ such that

[TABLE]

Here $K_{\varphi}^{V}$ denotes the unique ${\mathcal{K}}$-harmonic function in $V$, continuous up to $\partial V$ and such that $K_{\varphi}^{V}|_{\partial V}=\varphi.$

From this Claim our assertion follows thanks to Corollary 6.4.9 in [BLU07].

If $z\in\Omega\times]0,T_{0}[$ or if $z\in\Omega\times]0,T[$, the Claim is satisfied since $u$ is ${\mathcal{K}}$-superharmonic both in $\Omega\times]0,T_{0}[$ and in $\Omega\times]0,T[$. Then it remains to prove the Claim for every point $\zeta=(\xi,T_{0}),\xi\in\Omega.$ Let $B_{\rho}=(V)$ be a basis of ${\mathcal{K}}$-regular neighborhoods of $\zeta$ such that $\overline{V}\subseteq{\mathcal{O}}.$ Let $\varphi\in C(\partial V,{\mathbb{R}}),\varphi\leq u|_{\partial V}.$ Then $u-K_{\varphi}^{V}$ is ${\mathcal{K}}$-superharmonic in $\Omega\times]0,T_{0}[$ and

[TABLE]

Therefore, by Proposition 2.2,

[TABLE]

As a consequence, keeping in mind assumption (2.4),

[TABLE]

that is,

[TABLE]

This completes the proof.∎

3. Some preliminary results

The proof of our main theorem rests on the following two lemmata.

Lemma 3.1**.**

Let $\Omega$ be a bounded open set such that $\overline{\Omega}\subseteq X$, and let ${\mathcal{O}}:=\Omega\times]0,T[$, $T\in{\mathbb{R}},T>0.$ Let $\varphi:\partial{\mathcal{O}}\longrightarrow{\mathbb{R}}$ be upper semicontinuous and such that $t\longmapsto\varphi(x,t)$ is monotone decreasing, $\forall x\in\partial\Omega$ and

[TABLE]

Then, the Perron solution $K_{\varphi}^{\mathcal{O}}$ is monotone decreasing w.r.t. the variable $t$: more precisely

[TABLE]

Proof.

For every fixed $\delta\in]0,T[$ let us define

[TABLE]

It is enough to prove that $h\geq 0$ in ${\mathcal{O}}_{\delta}:=\Omega\times]0,T-\delta[.$ To this end we show that, for every $u\in\overline{\mathcal{U}}^{\mathcal{O}}_{\varphi}$ and $v\in\underline{\mathcal{U}}^{\mathcal{O}}_{\varphi}$, the function

[TABLE]

is nonnegative in ${\mathcal{O}}_{\delta}$. Now, we have:

• $(a)$

$w$ is ${\mathcal{K}}$-superharmonic in ${\mathcal{O}}_{\delta}$, since $u\in\overline{\mathcal{K}}({\mathcal{O}})$ and $(x,t)\longmapsto v(x,t+\delta)$ is ${\mathcal{K}}$-subharmonic in ${\mathcal{O}}_{\delta}$ being $v\in\underline{\mathcal{K}}({\mathcal{O}})$ and ${\mathscr{L}}$ translation invariant in the variable $t$.

• $(b)$

For every $\overline{x}\in\Omega,$

[TABLE]

We remark that $v\leq M$ in ${\mathcal{O}}$ since $v$ is ${\mathcal{K}}$-subharmonic and

[TABLE]

Here we use the maximum principle for subharmonic functions.

• $(c)$

For every $\zeta=(\xi,\tau)$, $\xi\in\partial\Omega,0<\tau<T-\delta,$

[TABLE]

by hypotesis.

From $(a)$, $(b)$ and $(c)$ and the minimum principle for superharmonic functions we get

[TABLE]

This completes the proof. ∎

With Lemma 3.1 at hand we can easily prove the following key result for our main theorem.

Lemma 3.2**.**

Let $\Omega$ be a bounded open set such that $\overline{\Omega}\subseteq X,$ and let ${\mathcal{O}}:=\Omega\times]0,T[$, $T\in{\mathbb{R}}$, $T>0.$ Let $z_{0}=(x_{0},t_{0})\in\partial\Omega\times]0,T[$ be a ${\mathscr{L}}$-regular boundary point.

Then there exists a function $b\in{\mathcal{K}}({\mathcal{O}})$ such that

• $(i)$

$b$* is an ${\mathscr{L}}$-barrier for ${\mathcal{O}}$ at $z_{0}$;*

• $(ii)$

$t\longmapsto b(x,t)$* is monotone decreasing for every fixed $x\in\Omega.$*

Proof.

Let $Y$ be a bounded open set such that $\overline{\Omega}\subseteq Y\subseteq\overline{Y}\subseteq X$ and let $x_{0}\in\Omega$. By Lemma 2.1 there exists a function $h\in C^{\infty}(Y,{\mathbb{R}})$ such that

• $(a)$

$h(x_{0})=0$ and $h(x)>0\quad\forall x\neq x_{0}.$

• $(b)$

${\mathscr{L}_{0}}h>0$ in $\Omega$.

For a fixed $\delta\in]0,T_{0}[$ let us define

[TABLE]

where $M=\sup_{\overline{\Omega}}h.$

This function is ${\mathscr{L}}$-superharmonic in ${\mathcal{O}}_{1}:=\Omega\times]0,\delta[$ and in ${\mathcal{O}}_{2}:=\Omega\times]\delta,T[$ since

[TABLE]

On the other hand,

[TABLE]

Then, by Proposition 2.3,

[TABLE]

Moreover,

[TABLE]

for every fixed $x\in\overline{\Omega}.$

Let us now put

[TABLE]

which is well defined and ${\mathcal{K}}$-harmonic in ${\mathcal{O}}$, since $\widehat{h}|_{\partial{\mathcal{O}}}$ is bounded and upper semicontinuous.

Moreover, by Lemma 3.1, $t\longmapsto b(x,t)$ is monotone decreasing for every fixed $x\in\Omega.$

It remains to show that $b$ is an ${\mathscr{L}}$-barrier for ${\mathcal{O}}$ at $z_{0}$. To this end we first remark that

[TABLE]

so that

[TABLE]

This implies $b>0$ in ${\mathcal{O}}$ since $\widehat{h}$ is strictly positive.

On the other hand, since $\widehat{h}|_{\partial{\mathcal{O}}}$ is continuous in a neighborhood of $z_{0}$, and $z_{0}$ is ${\mathscr{L}}$-regular for ${\mathcal{O}}$,

[TABLE]

This completes the proof.∎

4. Proof of Theorem 1.1

Let us keep the notation of Theorem 1.1 and split the proof in two steps.

• (1)

If $x_{0}\in\partial\Omega$ is ${\mathscr{L}}_{0}$-regular for $\Omega$, then $z=(x_{0},t_{0})$ is ${\mathscr{L}}$-regular for ${\mathcal{O}}.$

Indeed, the ${\mathscr{L}_{0}}$-regularity of $x_{0}$ implies the existence of a ${\mathscr{L}}_{0}$-harmonic barrier for $\Omega$ at $x_{0}$, i.e. a function $b_{0}\in{\mathcal{K}}(\Omega)$ such that

[TABLE]

It follows that

[TABLE]

is ${\mathscr{L}}$-harmonic in ${\mathcal{O}}$ (${\mathscr{L}}\widehat{b}={\mathscr{L}_{0}}b_{0}=0$). Moreover,

[TABLE]

Hence, $\widehat{b}$ is an ${\mathscr{L}}$-barrier function for ${\mathcal{O}}$ at $z_{0}$ and, as a consequence, $z_{0}$ is ${\mathscr{L}}$-regular for ${\mathcal{O}}$.

• (2)

If $z=(x_{0},t_{0})$, $x_{0}\in\Omega,0<t_{0}<T,$ is ${\mathscr{L}}$-regular for ${\mathcal{O}}$, then $x_{0}$ is ${\mathscr{L}_{0}}$-regular for $\Omega$.

Indeed, by Lemma 3.2, there exists a function $b\in{\mathcal{K}}({\mathcal{O}})$ such that $b>0,$  $b(z)\longrightarrow 0$ as $z\longrightarrow z_{0}$ and

[TABLE]

It follows that, letting $b_{0}(x)=b(x,t_{0}),$

[TABLE]

Hence, $b_{0}$ is ${\mathscr{L}_{0}}$-superharmonic in $\Omega.$ Moreover, $b_{0}>0$ in $\Omega$ and

[TABLE]

Therefore, $b_{0}$ is an ${\mathscr{L}}$-barrier for $\Omega$ at $x_{0}$ , and $x_{0}$ is ${\mathscr{L}}_{0}$-regular.

5. An application to degenerate Ornstein–Uhlenbeck operators

In ${\mathbb{R}}^{N}$ let us consider the partial differential operator

[TABLE]

where $A=(a_{ij})_{i,j=1,\dots,N}$ and $B=(b_{ij})_{i,j=1,\dots,N}$ are $N\times N$ real constant matrices, $x=(x_{1},\ldots,x_{N})$ is the point of ${\mathbb{R}}^{N},$ $\mathrm{div}$, $\nabla$ and $\langle\ ,\ \rangle$ denote the divergence, the Euclidean gradient and the inner product in ${\mathbb{R}}^{N}$, respectively.

We suppose that the matrix $A$ is symmetric, positive semidefinite and that it assumes the following block form

[TABLE]

$A_{0}$ being a $p_{0}\times p_{0}$ strictly positive definite matrix with $1\leq p_{0}\leq N$. Moreover, we assume the matrix $B$ to be of the following type

[TABLE]

where $B_{j}$ is a $p_{j-1}\times p_{j}$ block with rank $p_{j}$ ($j=1,2,...,r$), $p_{0}\geq p_{1}\geq...\geq p_{r}\geq 1$ and $p_{0}+p_{1}+...+p_{r}=N$.

Finally, letting

[TABLE]

we assume that the following condition is satisfied

[TABLE]

As it is quite well known this condition implies the hypoellipticity of $L$, see [LP94]. In that paper it is proved that the evolution counterpart of $L_{0}$, i.e. the operator

[TABLE]

is left translation invariant and homogeneous of degree two on the homogeneous group

[TABLE]

with composition law $\circ$ defined as follows

[TABLE]

and dilation $\delta_{\lambda},\lambda>0,$ of this kind

[TABLE]

where $x^{(p_{i})}\in{\mathbb{R}}^{p_{i}},\ i=0,\ldots,r.$

The natural number $q:=Q+2$, with

[TABLE]

is the homogenous dimension of $\mathbb{K}.$ In what follows we will write

[TABLE]

where,

[TABLE]

Obviously, $(D_{\lambda})_{\lambda>0}$ is a group of dilations in ${{\mathbb{R}}^{{N}}}$. The natural number $Q$ in (5.3) is the homogeneous dimension of ${{\mathbb{R}}^{{N}}}$ w.r.t. the group $(D_{\lambda})_{\lambda>0}$.

The operator $L$ has a fundamental solution $\Gamma$ given by

[TABLE]

where $\circ$ is the composition law in $\mathbb{K}$, $z^{-1}$ denotes the opposite of $z$ in $\mathbb{K}$ and, for a suitable $C_{Q}>0,$

[TABLE]

where,

[TABLE]

see again [LP94].

It is quite easy to recognise that our Tikhonov-type theorem applies to the operators $L_{0}$ and $L$. Hence, if $\Omega$ is a bounded open subset of ${\mathbb{R}}^{N}$, $x_{0}\in\partial\Omega$ and $t_{0}\in]-T,T[,T>0,$ we have:

[TABLE]

[TABLE]

[TABLE]

On the other hand, in [KLT18, Corollary 1.3] it is proved that

[TABLE]

if, for a $\mu\in]0,1[,$ the following condition holds:

[TABLE]

where $\alpha(k)=k\log k$, $|\cdot|$ denotes the Lebesque measure in ${\mathbb{R}}^{N+1}$ and

[TABLE]

We express now this condition in a more explicit form. To this end we let

[TABLE]

Then,

[TABLE]

Hence, denoting for the sake of brevity,

[TABLE]

condition (5.4) is satisfied if

[TABLE]

On the other hand, for every $p\in\mathbb{N},$

[TABLE]

[TABLE]

[TABLE]

Then, since $\dfrac{d_{p+1}}{\nu^{\alpha(p)}}\longrightarrow 0$ as $p\rightarrow\infty$ (as we will see later) condition (5.6) is satisfied if

[TABLE]

Keeping in mind the very definition of $\Gamma$, we have that $A_{k}(x_{0})$ is equal to the following set

[TABLE]

whereby, with the change of variables $y:=x_{0}-E(|t|)x,$ $\tau=-t$, we get

[TABLE]

Here $R_{k}=(C_{Q}\mu^{\alpha(k)})^{\frac{2}{Q}}$ and $\Omega^{c}:={{\mathbb{R}}^{{N+1}}}\smallsetminus\Omega$.

Therefore,

[TABLE]

Hence, for a suitable dimensional constant $C_{Q}^{*}>0,$

[TABLE]

Then,

[TABLE]

since $0<\mu<1$ and $\alpha(p+1)-\alpha(p)=p\log\frac{p+1}{p}+\log{(p+1)}\longrightarrow\infty.$

We have completed the proof of the following criterion:

Let $L$ be the Ornstein–Uhlenbeck-type operator in (5.1) and let $\Omega\subseteq{{\mathbb{R}}^{{N}}}$ be a bounded open set. Then, a point $x_{0}\in\partial\Omega$ is $L$-regular for $\Omega$ if

[TABLE]

where $d_{k}(\Omega,x_{0}):=d_{k}$ is defined in (5.8).

We note that condition (5.9) holds if $\Omega$ satisfies the exterior cone-type condition introduced in [Kog19]. Geometric boundary regularity criteria for wide classes of hypoelliptic evolution operators are also established in [Man97], [LU10], [LTU17] and [Kog17].

Acknowledgment

The author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).