# Tagged Questions

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### $\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)}$ holds?

I want to find the relation of $p$ and $k$ such that the inequality $$\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)}$$ holds when r.h.s $<\infty$. Here $f$ ...
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### Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume ...
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### Interior $H^2$ regularity - using a textbook's identity to show an important estimate

This is yet another continuation of my previous question, and concerns the same long textbook proof which I asked in those two questions as well. This is now from page 331 of PDE by Evans, 2nd ...
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### Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
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### Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
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### Show that for all $(\tau, \xi) \in \mathbb R^{n+1}$ we have $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$

Show that, for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$ This is the exercise 7.4 in the book by Francois Treves. It is just a fundamental ...
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### Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
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### Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
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### Morrey's inequality

From PDE Evans, 2nd edition, page 281: Now \begin{align} \int_0^s \int_{\partial B(0,1)} |Du(x+tw)| \, dS(w) dt &=\int_0^s \int_{\partial B(x,t)} \frac{|Du(y)|}{t^{n-1}} \, dS(y) dt \\ ...
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### Question on proof in Evans PDE

This is on page 542 of Evans PDE book. The last inequality states that $$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$$ Where is this coming from? I think ...
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Let $\Omega \subset \subset \mathbb{R}^N$ have smooth boundary, $N \geqslant 2$ and $$\mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ( v) \mathrm{d} x := ... 2answers 261 views ### An inequality about the gradient of a harmonic function Let G a open and connected set. Consider a function z=2R^{-\alpha}v-v^2 with R that will be chosen suitably small, where v is a harmonic function in G, and satisfies$$|x|^\alpha\leqslant ...
Consider the eigenfunction $\varphi_R>0$ $$L\varphi_R=\lambda_R\varphi_R, \ \ \ in \ \ B_R,$$ and $$\varphi_R=0, \ \ \ in \ \ \partial B_R,$$ where $L$ is a elliptic operator and $\lambda_R$ is the ...
### $W_0^{1,p}$ norm bounded by norm of Laplacian
Let $f\in W_0^{1,p}(U)$, for $U$ a bounded domain and $p < n/(n-1)$. I am trying to prove that there is an inequality of the form $$\|f\|_{W^{1,p}} \leq C \int_{\Omega} |\Delta f|$$ where ...