Examine whether a function in Sobolev spaces is smooth (C∞) on a bounded Lipschitz domain and determine bounds for its smoothness | Step-by-Step Solution

Hello! Great to see you diving into Sobolev spaces - this is a beautiful area where analysis meets geometry!

What We're Solving:

We're investigating whether functions in Sobolev spaces H^k(Ω) are automatically smooth (C^∞), and what bounds we can establish for their smoothness on a bounded Lipschitz domain Ω ⊂ ℝⁿ.

The Approach:

This problem is all about understanding the embedding theorems - these tell us when functions with certain "weak" regularity (measured by Sobolev norms) actually have "classical" smoothness properties. Think of it as asking: "If I know a function has k weak derivatives in L², what can I say about its pointwise behavior?"

Step-by-Step Solution:

Step 1: Understand what H^k(Ω) means

• H^k(Ω) contains functions whose weak derivatives up to order k are in L²(Ω)
• This gives us "integrated" control over derivatives, but not pointwise control yet
• The key insight: more Sobolev regularity can translate to classical smoothness!

Step 2: Apply the Sobolev Embedding Theorem The critical relationship depends on how k compares to n/2:

• Case 1: If k > n/2 + m for integer m ≥ 0, then H^k(Ω) ↪ C^m(Ω̄)
• Case 2: If k = n/2 + m, we get embeddings into Hölder spaces
• Case 3: If k < n/2, we don't get continuity

Step 3: Address the C^∞ question Here's the key insight: No finite Sobolev space H^k gives C^∞!

• C^∞ functions need ALL derivatives to exist and be continuous
• H^k only controls derivatives up to order k
• However, if f ∈ H^k for ALL k, then f ∈ C^∞

Step 4: Establish smoothness bounds For a bounded Lipschitz domain:

• The embedding constants depend on the domain's geometry
• Lipschitz boundary is "nice enough" for standard embedding theorems
• We get: ||f||_{C^m(Ω̄)} ≤ C||f||_{H^k(Ω)} when k > n/2 + m

Step 5: Consider the boundary behavior On Lipschitz domains:

• Interior regularity follows standard theory
• Boundary regularity is more delicate but still manageable
• The Lipschitz condition prevents "bad" boundary behavior

The Answer:

• 1. Smoothness: A function f ∈ H^k(Ω) is in C^m(Ω̄) if and only if k > n/2 + m
• 2. Not automatically C^∞: No finite k makes H^k ⊂ C^∞
• 3. Bounds: ||f||_{C^m(Ω̄)} ≤ C(Ω,n,k,m)||f||_{H^k(Ω)} when the embedding holds
• 4. Lipschitz domains: The bounded Lipschitz condition ensures these embeddings work with controlled constants

Memory Tip:

Remember the "magic number" n/2! When your Sobolev index k exceeds n/2, you start getting continuity. Think: "In higher dimensions, you need more Sobolev regularity to overcome the curse of dimensionality and achieve the same classical smoothness."

The beautiful thing here is how this connects weak (integral) control of derivatives to strong (pointwise) regularity - it's analysis at its finest! Keep exploring these connections - they're fundamental to PDEs and geometric analysis.