Get response for this dialogue: DIALOG:
What is Weyl's lemma?
- Weyl's lemma (Laplace equation) In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution.
- Are there any other interesting aspects about this article?
- This contrasts with the wave equation, for example, which has weak solutions that are not smooth solutions.
- Is there anything else noteable about the lemma?
- Weyl's lemma is a special case of elliptic or hypoelliptic regularity.
- Can you tell me what the equation is?
- Let formula_1 be an open subset of formula_2-dimensional Euclidean space formula_3, and let formula_4 denote the usual Laplace operator.
- What does it mean?
- Weyl's lemma states that if a locally integrable function formula_5 is a weak solution of Laplace's equation, in the sense that  for every smooth test function formula_7 with compact support, then (up to redefinition on a set of measure zero) formula_8 is smooth and satisfies formula_9 pointwise in formula_1.
- Is there anything else important?
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This result implies the interior regularity of harmonic functions in formula_1, but it does not say anything about their regularity on the boundary formula_12. To prove Weyl's lemma, one convolves the function formula_13 with an appropriate mollifier formula_14 and shows that the mollification formula_15 satisfies Laplace's equation, which implies that formula_16 has the mean value property.
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Who was Jarom ⁇ r, Duke of Bohemia?
- Jaromír, Duke of Bohemia  Jaromír (died 4 November 1035), a member of the Přemyslid dynasty, was Duke of Bohemia, in 1003, from 1004 to 1012, and again from 1034 to 1035.
- Who were his parents?
- He was the second son of Duke Boleslaus II the Pious (d. 999).
- What about his mother?
- His mother may have been either one of his father's two wives: Adiva or Emma of Mělník.
- Did he have any siblings?
- In 1002, Jaromír rebelled against the rule of his elder brother Boleslaus III, who had him castrated and expelled with his mother and his brother Oldřich to the Bavarian court at Regensburg.
- Are there any other interesting aspects about this article?
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Nevertheless, Boleslaus was unable to secure the Prague throne, as he was deposed by the Bohemian nobility and the rule was subsequently taken by his Přemyslid cousin Vladivoj, backed by the Polish duke Bolesław I the Brave.
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What is the scientific name of Notomulciber trimaculatus?
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Notomulciber trimaculatus Notomulciber trimaculatus is a species of beetle in the family Cerambycidae.
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What is the purpose of a Scene shop?
- Scene shop A scenery shop or scene shop is a specialized workshop found in many medium or large theaters, as well as many educational theatre settings.
- What is done in the scene shop?
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The primary function of a scene shop is to fabricate and assemble the flats, platforms, scenery wagons, and other scenic (set) pieces required for a performance.