Page of handwritten math problems and solutions.

To solve the problem, we need to carefully analyze the given conditions and use them to derive the required results. Let's break it down step by step.

Problem Setup

We are given a sequence of functions \( u_n \) in \( W^{1,p}(\Omega) \), where \( \Omega \) is an open subset of \( \mathbb{R}^N \). The sequence satisfies the following properties:

1. Uniform Boundedness in \( W^{1,p}(\Omega) \):
\[
\|u_n\|_{W^{1,p}(\Omega)} \leq C_0,
\]
for some constant \( C_0 > 0 \).

2. Pointwise Convergence:
\[
u_n(x) \to u(x) \quad \text{for almost every } x \in \Omega.
\]

3. Convergence of Gradients in Measure:
\[
Du_n \rightharpoonup^* \mu \quad \text{weakly-* in } \mathcal{M}(\Omega; \mathbb{R}^N),
\]
where \( \mu \) is a Radon measure on \( \Omega \).

4. Additional Regularity Condition:
\[
\int_\Omega |Du_n|^p \varphi \, dx \to \int_\Omega d\mu \varphi \quad \text{for all } \varphi \in C_c^\infty(\Omega).
\]

5. Decomposition of \( \mu \):
\[
\mu = Du + \nu,
\]
where \( Du \) is the weak derivative of \( u \) (which exists since \( u \in W^{1,p}(\Omega) \)) and \( \nu \) is a non-negative singular measure with respect to the Lebesgue measure.

6. Goal:
Show that:
\[
\int_\Omega |Du|^p \, dx \leq \liminf_{n \to \infty} \int_\Omega |Du_n|^p \, dx.
\]

---

Step-by-Step Solution

#### Step 1: Weak Convergence of Gradients
From the given conditions, we know that \( Du_n \rightharpoonup^* \mu \) weakly-* in the space of Radon measures \( \mathcal{M}(\Omega; \mathbb{R}^N) \). This implies that for any test function \( \varphi \in C_c^\infty(\Omega) \),
\[
\int_\Omega Du_n \cdot \nabla \varphi \, dx \to \int_\Omega d\mu \cdot \nabla \varphi.
\]
Since \( u_n \to u \) almost everywhere and \( u_n \) is bounded in \( W^{1,p}(\Omega) \), it follows that \( u \in W^{1,p}(\Omega) \) and \( Du_n \rightharpoonup Du \) weakly in \( L^p(\Omega; \mathbb{R}^N) \).

#### Step 2: Decomposition of \( \mu \)
The measure \( \mu \) can be decomposed as:
\[
\mu = Du + \nu,
\]
where \( Du \) is the absolutely continuous part of \( \mu \) with respect to the Lebesgue measure, and \( \nu \) is the singular part, which is non-negative.

#### Step 3: Convergence of \( |Du_n|^p \)
We are given that:
\[
\int_\Omega |Du_n|^p \varphi \, dx \to \int_\Omega d\mu \varphi \quad \text{for all } \varphi \in C_c^\infty(\Omega).
\]
This implies that \( |Du_n|^p \rightharpoonup^* \mu_p \) weakly-* in the sense of measures, where \( \mu_p \) is a Radon measure defined by:
\[
\int_\Omega \varphi \, d\mu_p = \lim_{n \to \infty} \int_\Omega |Du_n|^p \varphi \, dx.
\]

#### Step 4: Application of Lower Semicontinuity
By the lower semicontinuity of the \( L^p \)-norm under weak convergence, we have:
\[
\int_\Omega |Du|^p \, dx \leq \liminf_{n \to \infty} \int_\Omega |Du_n|^p \, dx.
\]
This is a standard result in functional analysis, which states that if \( f_n \rightharpoonup f \) weakly in \( L^p \), then:
\[
\|f\|_{L^p} \leq \liminf_{n \to \infty} \|f_n\|_{L^p}.
\]

#### Step 5: Conclusion
Combining the above results, we conclude that:
\[
\int_\Omega |Du|^p \, dx \leq \liminf_{n \to \infty} \int_\Omega |Du_n|^p \, dx.
\]

Thus, the final answer is:
\[
\boxed{\int_\Omega |Du|^p \, dx \leq \liminf_{n \to \infty} \int_\Omega |Du_n|^p \, dx.}
\]