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To address this problem, we need to define what it means for \( u \) to be a weak solution of the Poisson equation with mixed Dirichlet-Neumann boundary conditions and discuss the existence and uniqueness of these solutions.

### Weak Solution Definition

Consider the Poisson equation with mixed boundary conditions:

\[
\begin{cases}
-\Delta u = f & \text{in } U \\
u = 0 & \text{on } \Gamma_1 \\
\frac{\partial u}{\partial \nu} = 0 & \text{on } \Gamma_2,
\end{cases}
\]

where \( U \) is a connected domain, \( \Gamma_1 \) and \( \Gamma_2 \) are disjoint, closed sets forming the boundary \( \partial U \), \( \nu \) is the outward normal to \( \partial U \), and \( f \) is a given function.

A function \( u \in H^1(U) \) (the Sobolev space of functions with square-integrable first derivatives) is called a weak solution if it satisfies the following integral form for all test functions \( v \in H^1(U) \) that vanish on \( \Gamma_1 \):

\[
\int_U \nabla u \cdot \nabla v \, dx = \int_U f v \, dx.
\]

The boundary conditions are incorporated as follows:
- \( u = 0 \) on \( \Gamma_1 \) is explicitly required.
- The Neumann condition \( \frac{\partial u}{\partial \nu} = 0 \) on \( \Gamma_2 \) is naturally included in the weak formulation through the variational integral, where the boundary term vanishes.

### Existence and Uniqueness

**Existence**: The existence of a weak solution can be established using the Lax-Milgram theorem. The key steps are:

1. **Formulating the Problem**: Rewrite the Poisson equation as a variational problem:
   \[
   a(u, v) = \int_U \nabla u \cdot \nabla v \, dx,
   \]
   where \( a(u, v) \) is a bilinear form that is coercive and continuous on the appropriate Sobolev space.

2. **Lax-Milgram Theorem**: Apply the Lax-Milgram theorem, which guarantees the existence of a unique weak solution given that the bilinear form \( a(u, v) \) is coercive (bounded below by a positive constant) and continuous, and the linear functional \( \ell(v) = \int_U f v \, dx \) is continuous on \( H^1(U) \).

**Uniqueness**: The uniqueness of the weak solution follows directly from the coercivity of the bilinear form. Specifically, if there are two solutions \( u_1 \) and \( u_2 \), their difference \( w = u_1 - u_2 \) satisfies:

\[
a(w, v) = 0 \quad \forall v \in H^1(U).
\]

By choosing \( v = w \), we can show that \( w = 0 \), hence \( u_1 = u_2 \).

**Summary**: The weak solution \( u \in H^1(U) \) exists and is unique under the given mixed Dirichlet-Neumann boundary conditions if the functional \( \int_U f v \, dx \) is continuous and the bilinear form \( \int_U \nabla u \cdot \nabla v \, dx \) is coercive and continuous.

Would you like further details on the steps for the Lax-Milgram theorem or more information on the Sobolev spaces involved?

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Here are some related questions you might consider:
1. How does the Lax-Milgram theorem ensure the existence of a weak solution?
2. What are the properties of Sobolev spaces used in this context?
3. How do we handle different types of boundary conditions in weak formulations?
4. Can you show the steps for proving coercivity of the bilinear form?
5. How does the concept of weak derivatives apply to weak solutions?

**Tip:** When solving PDEs, always verify if the problem fits into the framework of existing theorems like Lax-Milgram to simplify proving existence and uniqueness.

Explore the definition, existence, and uniqueness of weak solutions for the Poisson equation with mixed Dirichlet-Neumann boundary conditions. This detailed explanation covers Sobolev spaces, variational formulations, and the application of the Lax-Milgram theorem. Ideal for advanced undergraduate and graduate students.

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