A discussion on the solvability of Dirichlet problem

Last Friday, I’ve had a pleasant discussion with Professor Lions (here) on my recent paper (here). PDF is also available (here).

The main topic was the existence of the viscosity solution (here for Definition) on a given PDE with Dirichlet boundary, and I told him please stop me anytime you think it’s a garbage.

We started with three toy problems for a motivation. Consider the following three ODEs with a fixed domain {O = (-1, 1)}.

\displaystyle   u' + u - 1 = 0, \hbox{ on } O , \quad u = 0, \hbox{ on } \partial O. \ \ \ \ \ (1)

\displaystyle   |u'| + u - 1 = 0, \hbox{ on } O , \quad u = 0, \hbox{ on } \partial O. \ \ \ \ \ (2)

and

\displaystyle   |u'| + u + 1 = 0, \hbox{ on } O , \quad u = 0, \hbox{ on } \partial O. \ \ \ \ \ (3)

By using the sufficient condition derived by Example 4.6 of [CIL92] (User’s Guide to Viscosity Solutions) together with some elementary analysis, the answer to the existence of the above three equations shall be

No, Yes, and No,

respectively. The main idea is to construct a sub and supersolutions, and then apply Perron’s method. However, as it is mentioned in Example 4.6 of [CIL92], it leaves an open question for the existence of a sub and supersolution in general.

Next, we take a look at two equations, which may not be a direct consequence of Example 4.6.

Example 1

Let {\alpha \in (0,1)} be a constant and {O = B_{1} \subset \mathbb R^{d}} be a unit open ball.

  1. (Linear) Given a constant {a\in \mathbb R^{d}}, answer the solvability of

    \displaystyle   - a \cdot \nabla u + (- \Delta)^{\alpha/2} u + u - 1 = 0, \hbox{ on } O, \quad u = 0, \hbox{ on } O^{c}. \ \ \ \ \ (4)

  2. (Nonlinear) Answer the solvability of

    \displaystyle   | \nabla u | + (- \Delta)^{\alpha/2} u + u - 1 = 0, \hbox{ on } O, \quad u = 0, \hbox{ on } O^{c}. \ \ \ \ \ (5)

-END

If {\alpha \in [1, 2]} in the above example, then the answers are Yes for both, according to the paper [BCI08] (here). But if {\alpha <1}, we will answer the question in this below.

To proceed, let’s first recall some well known facts and terminologies:

  1. Let {\mathbb D} be the collection of RCLL processes {\omega: [0, \infty) \mapsto \mathbb R^{d}} equipped with Skorohod metric;
  2. We fix {\alpha \in (0, 1)} in the rest of the notes. For {m:\mathbb R^{d} \mapsto \mathbb R^{d}}, we denote

    \displaystyle \mathcal L^{m} := m \cdot \nabla - (- \Delta)^{\alpha/2}.

    If {m \in C^{0,1}(\mathbb R^{d}, \mathbb R^{d})}, it is well known that there exists a Feller process {X \sim \mathcal L^{m}.} We know that a Feller process is strong Markovian and takes its path in {\mathbb D} with probability one.

  3. Define exit times as the mappings {\tau, \bar \tau: \mathbb D \mapsto \mathbb R} by

    \displaystyle \tau(\omega) = \inf\{t >0: \omega_{t} \in O^{c}\}, \quad \bar \tau (\omega) = \inf\{t > 0: \omega_{t} \in \bar O^{c}\}.

  4. We say {\partial O} is regular w.r.t. {\mathcal L}, and denoted by {\partial O \in Reg(\mathcal L)}, if there exists a strong Markov {X\sim \mathcal L} satisfying

    \displaystyle \mathbb P^{x} \{\bar \tau(X) = 0\} = 1, \ \forall x\in \partial O.

Theorem 1 (Linear)

If

  1. {\ell \in C^{0,1}(\mathbb R^{d}, \mathbb R)},
  2. {m \in C^{0,1}(\mathbb R^{d}, \mathbb R^{d})} is a given vector field such that {\partial O \in Reg(\mathcal L^{m})},

then there exists a unique solution of

\displaystyle   - \mathcal L^{m} u + u - \ell = 0, \hbox{ on } O, \quad u = 0, \hbox{ on } O^{c}. \ \ \ \ \ (6)

-END

Theorem 2 (Nonlinear)

Let {\Lambda = \{a\in \mathbb R^{d}: |a| \le 1\}}. If

  1. {\ell \in C^{0,1}(\mathbb R^{d}, \mathbb R)} is given {\ell \ge 0};
  2. {\exists m \in C^{0,1}(\mathbb R^{d}, \mathbb R^{d})} such that { \partial O \in Reg (\mathcal L^{m})};

then there exists a unique solution of

\displaystyle  \sup_{a\in \Lambda} - \mathcal L^{a} u + u - \ell = 0, \hbox{ on } O, \quad u = 0, \hbox{ on } O^{c}.

-END

The answer to Example 1 is given below now:

  1. (4) is solvable if and only if {a = 0}. Theorem 1 answers if direction, while an obvious reason answers only if part;
  2. (5) is solvable due to Theorem 2 with the choice {m = 0}.
  3. The above answers shall remain the same if {O } is merely an open bounded set satisfying exterior cone condition. (At this time, Lions said, it is certainly not a garbage.)

Proof: We sketch the proof of two theorems and more details are referred to the paper.

  1. Theorem 2 is a consequence of Perron’s method, given that Theorem 1 is true. We shall prove Theorem 1 in this below.
  2. Let {X\sim \mathcal L^{m}} of Theorem 1, and define

    \displaystyle v(x) = \mathbb E^{x} \Big[ \int_{0}^{\tau} e^{-s} \ell(X_{s}) ds \Big].

    Given that {v\in C(\bar O)}, then it is routine to show that {v} is solution of (6). It is remained to show {v\in C(\bar O)}.

  3. One can rewrite {v(x) = \mathbb E^{x} [ F(X, \tau) ]} with obvious choice of {F}. Now, we finish the proof of {v\in C(\bar O)} by checking the following three facts:
    1. {F: \mathbb D\times [0, \infty) \mapsto \mathbb R} is continuous;
    2. {\omega \mapsto \tau(\omega)} is continuous on {\Gamma};
    3. {\mathbb P^{x}(\Gamma) = 1}.

    In the above, {\Gamma} is the collection of {\omega \in \mathbb D} satisfying

    1. {\omega} is continuous at {\tau(\omega^{-})};
    2. {\tau(\omega) = \bar \tau (\omega)}.

\Box

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